{"id":286,"date":"2021-02-04T00:49:20","date_gmt":"2021-02-04T00:49:20","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=286"},"modified":"2022-03-11T21:55:36","modified_gmt":"2022-03-11T21:55:36","slug":"the-squeeze-theorem","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/the-squeeze-theorem\/","title":{"raw":"The Squeeze Theorem","rendered":"The Squeeze Theorem"},"content":{"raw":"
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Learning Outcomes<\/h3>\r\n
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  • Evaluate the limit of a function by using the squeeze theorem<\/span><\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<\/div>\r\n

    The techniques we have developed thus far work very well for algebraic functions, but we are still unable to evaluate limits of very basic trigonometric functions. The next theorem, called the squeeze theorem<\/strong>, proves very useful for establishing basic trigonometric limits. This theorem allows us to calculate limits by \u201csqueezing\u201d a function, with a limit at a point [latex]a[\/latex] that is unknown, between two functions having a common known limit at [latex]a[\/latex]. Figure 5 illustrates this idea.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"A Figure 5. The Squeeze Theorem applies when [latex]f(x)\\le g(x)\\le h(x)[\/latex] and [latex]\\underset{x\\to a}{\\lim}f(x)=\\underset{x\\to a}{\\lim}h(x)[\/latex].[\/caption]\r\n

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    The Squeeze Theorem<\/h3>\r\n\r\n
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    Let [latex]f(x), \\, g(x)[\/latex], and [latex]h(x)[\/latex] be defined for all [latex]x\\ne a[\/latex] over an open interval containing [latex]a[\/latex]. If<\/p>\r\n\r\n

    [latex]f(x)\\le g(x)\\le h(x)[\/latex]<\/div>\r\n \r\n
    <\/div>\r\n

    for all [latex]x\\ne a[\/latex] in an open interval containing [latex]a[\/latex] and<\/p>\r\n\r\n

    [latex]\\underset{x\\to a}{\\lim}f(x)=L=\\underset{x\\to a}{\\lim}h(x)[\/latex]<\/div>\r\n \r\n
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    where [latex]L[\/latex] is a real number, then [latex]\\underset{x\\to a}{\\lim}g(x)=L[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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    Example: Applying the Squeeze Theorem<\/h3>\r\n

    Apply the Squeeze Theorem to evaluate [latex]\\underset{x\\to 0}{\\lim}x \\cos x[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1170571654269\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571654269\"]\r\n\r\nBecause [latex]-1\\le \\cos x\\le 1[\/latex] for all [latex]x[\/latex], we have [latex]-|x|\\le x \\cos x\\le |x|[\/latex]. Since [latex]\\underset{x\\to 0}{\\lim}(-|x|)=0=\\underset{x\\to 0}{\\lim}|x|[\/latex], from the Squeeze Theorem, we obtain [latex]\\underset{x\\to 0}{\\lim}x \\cos x=0[\/latex]. The graphs of [latex]f(x)=-|x|, \\, g(x)=x \\cos x[\/latex], and [latex]h(x)=|x|[\/latex] are shown in Figure 6.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"312\"]\"The Figure 6. The graphs of \ud835\udc53(\ud835\udc65), \ud835\udc54(\ud835\udc65), and \u210e(\ud835\udc65) are shown around the point \ud835\udc65=0.[\/caption]\r\n\r\n \r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

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    Try It<\/h3>\r\n

    Use the Squeeze Theorem to evaluate [latex]\\underset{x\\to 0}{\\lim}x^2 \\sin \\dfrac{1}{x}[\/latex].<\/p>\r\n[reveal-answer q=\"28643309\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"28643309\"]\r\n

    Use the fact that [latex]-x^2\\le x^2 \\sin (\\frac{1}{x})\\le x^2[\/latex] to help you find two functions such that [latex]x^2 \\sin (\\frac{1}{x})[\/latex] is squeezed between them.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572560337\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572560337\"]\r\n

    0<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to the above Try It. [\/caption]\r\n\r\n