{"id":488,"date":"2021-02-04T15:31:01","date_gmt":"2021-02-04T15:31:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=488"},"modified":"2021-06-24T05:12:29","modified_gmt":"2021-06-24T05:12:29","slug":"problem-set-limits-at-infinity-and-asymptotes","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/problem-set-limits-at-infinity-and-asymptotes\/","title":{"raw":"Problem Set: Limits at Infinity and Asymptotes","rendered":"Problem Set: Limits at Infinity and Asymptotes"},"content":{"raw":"For the following exercises, examine the graphs. Identify where the vertical asymptotes are located.\r\n
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1.\u00a0<\/strong>\"The\r\n[reveal-answer q=\"fs-id1165042640653\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042640653\"][latex]x=1[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n
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2.\u00a0<\/strong>\"The<\/span><\/div>\r\n<\/div>\r\n
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3.\u00a0<\/strong>\"The\r\n[reveal-answer q=\"fs-id1165043197243\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043197243\"][latex]x=-1, \\, x=2[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n
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4.<\/strong>\u00a0\"The<\/span><\/div>\r\n<\/div>\r\n
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5.<\/strong>\u00a0\"The\r\n[reveal-answer q=\"fs-id1165043197340\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043197340\"][latex]x=0[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n

For the following functions [latex]f(x)[\/latex], determine whether there is an asymptote at [latex]x=a[\/latex]. Justify your answer without graphing on a calculator.<\/p>\r\n\r\n

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6.<\/strong> [latex]f(x)=\\dfrac{x+1}{x^2+5x+4}, \\, a=-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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7.<\/strong> [latex]f(x)=\\dfrac{x}{x-2}, \\, a=2[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043197490\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043197490\"]\r\n

Yes, there is a vertical asymptote<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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8.<\/strong> [latex]f(x)=(x+2)^{3\/2}, \\, a=-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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9.<\/strong> [latex]f(x)=(x-1)^{-1\/3}, \\, a=1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043197610\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043197610\"]\r\n

Yes, there is a vertical asymptote<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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10.<\/strong> [latex]f(x)=1+x^{-2\/5}, \\, a=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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For the following exercises, evaluate the limit.<\/span><\/h2>\r\n<\/div>\r\n
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11.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{1}{3x+6}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043377772\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043377772\"]\r\n

0<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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12.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{2x-5}{4x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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13.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{x^2-2x+5}{x+2}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043377884\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043377884\"]\r\n

[latex]\\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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14.<\/strong> [latex]\\underset{x\\to \u2212\\infty }{\\lim}\\dfrac{3x^3-2x}{x^2+2x+8}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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15.<\/strong> [latex]\\underset{x\\to \u2212\\infty }{\\lim}\\dfrac{x^4-4x^3+1}{2-2x^2-7x^4}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043378034\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043378034\"]\r\n

[latex]-\\frac{1}{7}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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16.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{3x}{\\sqrt{x^2+1}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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17.<\/strong> [latex]\\underset{x\\to \u2212\\infty }{\\lim}\\dfrac{\\sqrt{4x^2-1}}{x+2}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043216394\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043216394\"]\r\n

-2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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18.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{4x}{\\sqrt{x^2-1}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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19.<\/strong> [latex]\\underset{x\\to \u2212\\infty }{\\lim}\\dfrac{4x}{\\sqrt{x^2-1}}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043216501\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043216501\"]\r\n

-4<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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20.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{2\\sqrt{x}}{x-\\sqrt{x}+1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find the horizontal and vertical asymptotes.<\/p>\r\n\r\n

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21.<\/strong> [latex]f(x)=x-\\dfrac{9}{x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043216600\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043216600\"]\r\n

Horizontal: none, vertical: [latex]x=0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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22.<\/strong> [latex]f(x)=\\dfrac{1}{1-x^2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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23.<\/strong> [latex]f(x)=\\dfrac{x^3}{4-x^2}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043216720\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043216720\"]\r\n

Horizontal: none, vertical: [latex]x=\\pm 2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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24.<\/strong> [latex]f(x)=\\dfrac{x^2+3}{x^2+1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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25.<\/strong> [latex]f(x)= \\sin (x) \\sin (2x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043210340\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043210340\"]\r\n

Horizontal: none, vertical: none<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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26.<\/strong> [latex]f(x)= \\cos x+ \\cos (3x)+ \\cos (5x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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27.<\/strong> [latex]f(x)=\\dfrac{x \\sin (x)}{x^2-1}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043210457\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043210457\"]\r\n

Horizontal: [latex]y=0[\/latex], vertical: [latex]x=\\pm 1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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28.\u00a0<\/strong>[latex]f(x)=\\dfrac{x}{ \\sin (x)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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29.<\/strong> [latex]f(x)=\\dfrac{1}{x^3+x^2}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043210591\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043210591\"]\r\n

Horizontal: [latex]y=0[\/latex], vertical: [latex]x=0[\/latex] and [latex]x=-1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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30.<\/strong> [latex]f(x)=\\dfrac{1}{x-1}-2x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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31.<\/strong> [latex]f(x)=\\dfrac{x^3+1}{x^3-1}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043349244\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043349244\"]\r\n

Horizontal: [latex]y=1[\/latex], vertical: [latex]x=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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32.<\/strong> [latex]f(x)=\\dfrac{ \\sin x+ \\cos x}{ \\sin x- \\cos x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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33.<\/strong> [latex]f(x)=x- \\sin x[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043349378\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043349378\"]\r\n

Horizontal: none, vertical: none<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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34.<\/strong> [latex]f(x)=\\dfrac{1}{x}-\\sqrt{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, construct a function [latex]f(x)[\/latex] that has the given asymptotes.<\/p>\r\n\r\n

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35.<\/strong> [latex]x=1[\/latex] and [latex]y=2[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043349476\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043349476\"]\r\n

Answers will vary, for example: [latex]y=\\frac{2x}{x-1}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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36.<\/strong> [latex]x=1[\/latex] and [latex]y=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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37.<\/strong> [latex]y=4[\/latex] and [latex]x=-1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043349589\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043349589\"]\r\n

Answers will vary, for example: [latex]y=\\frac{4x}{x+1}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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38.<\/strong> [latex]x=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (40-44), graph the function on a graphing calculator on the window [latex]x=[-5,5][\/latex] and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit.<\/p>\r\n\r\n

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39. [T]\u00a0<\/strong>[latex]f(x)=\\dfrac{1}{x+10}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042709177\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042709177\"]\r\n

[latex]y=0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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40. [T]\u00a0<\/strong>[latex]f(x)=\\dfrac{x+1}{x^2+7x+6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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41. [T]\u00a0<\/strong>[latex]\\underset{x\\to \u2212\\infty }{\\lim} x^2+10x+25[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042709306\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042709306\"]\r\n

[latex]\\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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42. [T]\u00a0<\/strong>[latex]\\underset{x\\to \u2212\\infty }{\\lim}\\dfrac{x+2}{x^2+7x+6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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43. [T]\u00a0<\/strong>[latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{3x+2}{x+5}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042709433\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042709433\"]\r\n

[latex]y=3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (45-56), draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior.<\/p>\r\n\r\n

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44.<\/strong> [latex]y=3x^2+2x+4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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45.<\/strong> [latex]y=x^3-3x^2+4[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042558120\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042558120\"]\"The<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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46.<\/strong> [latex]y=\\dfrac{2x+1}{x^2+6x+5}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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47.<\/strong> [latex]y=\\dfrac{x^3+4x^2+3x}{3x+9}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042558253\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042558253\"]\"An<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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48.<\/strong> [latex]y=\\dfrac{x^2+x-2}{x^2-3x-4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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49.<\/strong> [latex]y=\\sqrt{x^2-5x+4}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042558376\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042558376\"]\"This<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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50.<\/strong> [latex]y=2x\\sqrt{16-x^2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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51.<\/strong> [latex]y=\\dfrac{ \\cos x}{x}[\/latex], on [latex]x=[-2\\pi ,2\\pi][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042558496\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042558496\"]\"This<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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52.<\/strong> [latex]y=e^x-x^3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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53.<\/strong> [latex]y=x \\tan x, \\, x=[\u2212\\pi ,\\pi][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043404067\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043404067\"]\"This<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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54.<\/strong> [latex]y=x \\ln (x), \\, x>0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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55.<\/strong> [latex]y=x^2 \\sin (x), \\, x=[-2\\pi ,2\\pi][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043404194\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043404194\"]\"This<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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56.<\/strong> For [latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] to have an asymptote at [latex]y=2[\/latex] then the polynomials [latex]P(x)[\/latex] and [latex]Q(x)[\/latex] must have what relation?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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57.<\/strong> For [latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] to have an asymptote at [latex]x=0[\/latex], then the polynomials [latex]P(x)[\/latex] and [latex]Q(x)[\/latex] must have what relation?<\/p>\r\n[reveal-answer q=\"fs-id1165043208543\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043208543\"]\r\n

The degree of [latex]Q(x)[\/latex] must be greater than the degree of [latex]P(x)[\/latex].<\/p>\r\n[latex]P(0)\\ne{0}[\/latex] and [latex]Q(x)=0[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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58.<\/strong> If [latex]f^{\\prime}(x)[\/latex] has asymptotes at [latex]y=3[\/latex] and [latex]x=1[\/latex], then [latex]f(x)[\/latex] has what asymptotes?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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59.<\/strong> Both [latex]f(x)=\\dfrac{1}{x-1}[\/latex] and [latex]g(x)=\\dfrac{1}{(x-1)^2}[\/latex] have asymptotes at [latex]x=1[\/latex] and [latex]y=0[\/latex]. What is the most obvious difference between these two functions?<\/p>\r\n[reveal-answer q=\"fs-id1165043208777\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043208777\"]\r\n

[latex]\\underset{x\\to 1^-}{\\lim} f(x)=-\\infty[\/latex] and [latex]\\underset{x\\to 1^-}{\\lim} g(x)=\\infty[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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60.<\/strong> True or false: Every ratio of polynomials has vertical asymptotes.<\/p>\r\n\r\n<\/div>\r\n<\/div>","rendered":"

For the following exercises, examine the graphs. Identify where the vertical asymptotes are located.<\/p>\n

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1.\u00a0<\/strong>\"The<\/p>\n
Show Solution<\/span><\/p>\n
[latex]x=1[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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2.\u00a0<\/strong>\"The<\/span><\/div>\n<\/div>\n
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3.\u00a0<\/strong>\"The<\/p>\n
Show Solution<\/span><\/p>\n
[latex]x=-1, \\, x=2[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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4.<\/strong>\u00a0\"The<\/span><\/div>\n<\/div>\n
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5.<\/strong>\u00a0\"The<\/p>\n
Show Solution<\/span><\/p>\n
[latex]x=0[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following functions [latex]f(x)[\/latex], determine whether there is an asymptote at [latex]x=a[\/latex]. Justify your answer without graphing on a calculator.<\/p>\n

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6.<\/strong> [latex]f(x)=\\dfrac{x+1}{x^2+5x+4}, \\, a=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n

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7.<\/strong> [latex]f(x)=\\dfrac{x}{x-2}, \\, a=2[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Yes, there is a vertical asymptote<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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8.<\/strong> [latex]f(x)=(x+2)^{3\/2}, \\, a=-2[\/latex]<\/p>\n<\/div>\n<\/div>\n

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9.<\/strong> [latex]f(x)=(x-1)^{-1\/3}, \\, a=1[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Yes, there is a vertical asymptote<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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10.<\/strong> [latex]f(x)=1+x^{-2\/5}, \\, a=1[\/latex]<\/p>\n<\/div>\n<\/div>\n

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For the following exercises, evaluate the limit.<\/span><\/h2>\n<\/div>\n
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11.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{1}{3x+6}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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12.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{2x-5}{4x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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13.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{x^2-2x+5}{x+2}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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14.<\/strong> [latex]\\underset{x\\to \u2212\\infty }{\\lim}\\dfrac{3x^3-2x}{x^2+2x+8}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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15.<\/strong> [latex]\\underset{x\\to \u2212\\infty }{\\lim}\\dfrac{x^4-4x^3+1}{2-2x^2-7x^4}[\/latex]<\/p>\n

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[latex]-\\frac{1}{7}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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16.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{3x}{\\sqrt{x^2+1}}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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17.<\/strong> [latex]\\underset{x\\to \u2212\\infty }{\\lim}\\dfrac{\\sqrt{4x^2-1}}{x+2}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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-2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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18.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{4x}{\\sqrt{x^2-1}}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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19.<\/strong> [latex]\\underset{x\\to \u2212\\infty }{\\lim}\\dfrac{4x}{\\sqrt{x^2-1}}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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-4<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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20.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{2\\sqrt{x}}{x-\\sqrt{x}+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises, find the horizontal and vertical asymptotes.<\/p>\n

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21.<\/strong> [latex]f(x)=x-\\dfrac{9}{x}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Horizontal: none, vertical: [latex]x=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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22.<\/strong> [latex]f(x)=\\dfrac{1}{1-x^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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23.<\/strong> [latex]f(x)=\\dfrac{x^3}{4-x^2}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Horizontal: none, vertical: [latex]x=\\pm 2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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24.<\/strong> [latex]f(x)=\\dfrac{x^2+3}{x^2+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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25.<\/strong> [latex]f(x)= \\sin (x) \\sin (2x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Horizontal: none, vertical: none<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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26.<\/strong> [latex]f(x)= \\cos x+ \\cos (3x)+ \\cos (5x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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27.<\/strong> [latex]f(x)=\\dfrac{x \\sin (x)}{x^2-1}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Horizontal: [latex]y=0[\/latex], vertical: [latex]x=\\pm 1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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28.\u00a0<\/strong>[latex]f(x)=\\dfrac{x}{ \\sin (x)}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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29.<\/strong> [latex]f(x)=\\dfrac{1}{x^3+x^2}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Horizontal: [latex]y=0[\/latex], vertical: [latex]x=0[\/latex] and [latex]x=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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30.<\/strong> [latex]f(x)=\\dfrac{1}{x-1}-2x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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31.<\/strong> [latex]f(x)=\\dfrac{x^3+1}{x^3-1}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Horizontal: [latex]y=1[\/latex], vertical: [latex]x=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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32.<\/strong> [latex]f(x)=\\dfrac{ \\sin x+ \\cos x}{ \\sin x- \\cos x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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33.<\/strong> [latex]f(x)=x- \\sin x[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Horizontal: none, vertical: none<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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34.<\/strong> [latex]f(x)=\\dfrac{1}{x}-\\sqrt{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises, construct a function [latex]f(x)[\/latex] that has the given asymptotes.<\/p>\n

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35.<\/strong> [latex]x=1[\/latex] and [latex]y=2[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Answers will vary, for example: [latex]y=\\frac{2x}{x-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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36.<\/strong> [latex]x=1[\/latex] and [latex]y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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37.<\/strong> [latex]y=4[\/latex] and [latex]x=-1[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Answers will vary, for example: [latex]y=\\frac{4x}{x+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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38.<\/strong> [latex]x=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises (40-44), graph the function on a graphing calculator on the window [latex]x=[-5,5][\/latex] and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit.<\/p>\n

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39. [T]\u00a0<\/strong>[latex]f(x)=\\dfrac{1}{x+10}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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40. [T]\u00a0<\/strong>[latex]f(x)=\\dfrac{x+1}{x^2+7x+6}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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41. [T]\u00a0<\/strong>[latex]\\underset{x\\to \u2212\\infty }{\\lim} x^2+10x+25[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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42. [T]\u00a0<\/strong>[latex]\\underset{x\\to \u2212\\infty }{\\lim}\\dfrac{x+2}{x^2+7x+6}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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43. [T]\u00a0<\/strong>[latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{3x+2}{x+5}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]y=3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (45-56), draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior.<\/p>\n

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44.<\/strong> [latex]y=3x^2+2x+4[\/latex]<\/p>\n<\/div>\n<\/div>\n

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45.<\/strong> [latex]y=x^3-3x^2+4[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"The<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n
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46.<\/strong> [latex]y=\\dfrac{2x+1}{x^2+6x+5}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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47.<\/strong> [latex]y=\\dfrac{x^3+4x^2+3x}{3x+9}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"An<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n
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48.<\/strong> [latex]y=\\dfrac{x^2+x-2}{x^2-3x-4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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49.<\/strong> [latex]y=\\sqrt{x^2-5x+4}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n
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50.<\/strong> [latex]y=2x\\sqrt{16-x^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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51.<\/strong> [latex]y=\\dfrac{ \\cos x}{x}[\/latex], on [latex]x=[-2\\pi ,2\\pi][\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n
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52.<\/strong> [latex]y=e^x-x^3[\/latex]<\/p>\n<\/div>\n<\/div>\n

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53.<\/strong> [latex]y=x \\tan x, \\, x=[\u2212\\pi ,\\pi][\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n
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54.<\/strong> [latex]y=x \\ln (x), \\, x>0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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55.<\/strong> [latex]y=x^2 \\sin (x), \\, x=[-2\\pi ,2\\pi][\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n
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56.<\/strong> For [latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] to have an asymptote at [latex]y=2[\/latex] then the polynomials [latex]P(x)[\/latex] and [latex]Q(x)[\/latex] must have what relation?<\/p>\n<\/div>\n<\/div>\n

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57.<\/strong> For [latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] to have an asymptote at [latex]x=0[\/latex], then the polynomials [latex]P(x)[\/latex] and [latex]Q(x)[\/latex] must have what relation?<\/p>\n

Show Solution<\/span><\/p>\n
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The degree of [latex]Q(x)[\/latex] must be greater than the degree of [latex]P(x)[\/latex].<\/p>\n

[latex]P(0)\\ne{0}[\/latex] and [latex]Q(x)=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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58.<\/strong> If [latex]f^{\\prime}(x)[\/latex] has asymptotes at [latex]y=3[\/latex] and [latex]x=1[\/latex], then [latex]f(x)[\/latex] has what asymptotes?<\/p>\n<\/div>\n<\/div>\n

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59.<\/strong> Both [latex]f(x)=\\dfrac{1}{x-1}[\/latex] and [latex]g(x)=\\dfrac{1}{(x-1)^2}[\/latex] have asymptotes at [latex]x=1[\/latex] and [latex]y=0[\/latex]. What is the most obvious difference between these two functions?<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\underset{x\\to 1^-}{\\lim} f(x)=-\\infty[\/latex] and [latex]\\underset{x\\to 1^-}{\\lim} g(x)=\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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60.<\/strong> True or false: Every ratio of polynomials has vertical asymptotes.<\/p>\n<\/div>\n<\/div>\n\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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