Problem Set: The Precise Definition of a Limit

In the following exercises (1-4), write the appropriate $\varepsilon$ – $\delta$ definition for each of the given statements.

1. $\underset{x\to a}{\lim}f(x)=N$

2. $\underset{t\to b}{\lim}g(t)=M$

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For every $\varepsilon >0$ , there exists a $\delta >0$ so that if $0<|t-b|<\delta$ , then $|g(t)-M|<\varepsilon$

3. $\underset{x\to c}{\lim}h(x)=L$

4. $\underset{x\to a}{\lim}\phi(x)=A$

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For every $\varepsilon >0$ , there exists a $\delta >0$ so that if $0<|x-a|<\delta$ , then $|\phi(x)-A|<\varepsilon$

The following graph of the function $f$ satisfies $\underset{x\to 2}{\lim}f(x)=2$ . In the following exercises (5-6), determine a value of $\delta >0$ that satisfies each statement.

A function drawn in quadrant one for x > 0. It is an increasing concave up function, with points approximately (0,0), (1, .5), (2,2), and (3,4).

5. If $0<|x-2|<\delta$ , then $|f(x)-2|<1$ .

6. If $0<|x-2|<\delta$ , then $|f(x)-2|<0.5$ .

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$\delta \le 0.25$

The following graph of the function $f$ satisfies $\underset{x\to 3}{\lim}f(x)=-1$ . In the following exercises (7-8), determine a value of $\delta >0$ that satisfies each statement.

A graph of a decreasing linear function, with points (0,2), (1,1), (2,0), (3,-1), (4,-2), and so on for x >= 0.

7. If $0<|x-3|<\delta$ , then $|f(x)+1|<1$ .

8. If $0<|x-3|<\delta$ , then $|f(x)+1|<2$ .

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$\delta \le 2$

The following graph of the function $f$ satisfies $\underset{x\to 3}{\lim}f(x)=2$ . In the following exercises (9-10), for each value of $\varepsilon$ , find a value of $\delta >0$ such that the precise definition of limit holds true.

A graph of an increasing linear function intersecting the x axis at about (2.25, 0) and going through the points (3,2) and, approximately, (1,-5) and (4,5).

9. $\varepsilon =1.5$

10. $\varepsilon =3$

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$\delta \le 1$

In the following exercises (11-12), use a graphing calculator to find a number $\delta$ such that the statements hold true.

11. [T] $|\sin (2x)-\frac{1}{2}|<0.1$ , whenever $|x-\frac{\pi}{12}|<\delta$

12. [T] $|\sqrt{x-4}-2|<0.1$ , whenever $|x-8|<\delta$

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$\delta <0.3900$

In the following exercises (13-17), use the precise definition of limit to prove the given limits.

13. $\underset{x\to 2}{\lim}(5x+8)=18$

14. $\underset{x\to 3}{\lim}\dfrac{x^2-9}{x-3}=6$

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Let $\delta =\varepsilon$ . If $0<|x-3|<\varepsilon$ , then $|x+3-6|=|x-3|<\varepsilon$ .

15. $\underset{x\to 2}{\lim}\dfrac{2x^2-3x-2}{x-2}=5$

16. $\underset{x\to 0}{\lim}x^4=0$

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Let $\delta =\sqrt[4]{\varepsilon}$ . If $0<|x|<\sqrt[4]{\varepsilon}$ , then $|x^4|=x^4<\varepsilon$ .

17. $\underset{x\to 2}{\lim}(x^2+2x)=8$

In the following exercises (18-20), use the precise definition of limit to prove the given one-sided limits.

18. $\underset{x\to 5^-}{\lim}\sqrt{5-x}=0$

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Let $\delta =\varepsilon^2$ . If [latex]5-\varepsilon^2

19. $\underset{x\to 0^+}{\lim}f(x)=-2$ , where $f(x)=\begin{cases} 8x-3 & \text{ if } \, x<0 \\ 4x-2 & \text{ if } \, x \ge 0 \end{cases}$

20. $\underset{x\to 1^-}{\lim}f(x)=3$ , where $f(x)=\begin{cases} 5x-2 & \text{ if } \, x < 1 \\ 7x-1 & \text{ if } x \ge 1 \end{cases}$

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Let $\delta =\varepsilon/5$ . If [latex]1-\varepsilon/5

In the following exercises (21-23), use the precise definition of limit to prove the given infinite limits.

21. $\underset{x\to 0}{\lim}\dfrac{1}{x^2}=\infty$

22. $\underset{x\to -1}{\lim}\dfrac{3}{(x+1)^2}=\infty$

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Let $\delta =\sqrt{\frac{3}{N}}$ . If $0<|x+1|<\sqrt{\frac{3}{N}}$ , then $f(x)=\frac{3}{(x+1)^2}>N$ .

23. $\underset{x\to 2}{\lim}-\dfrac{1}{(x-2)^2}=−\infty$

24. An engineer is using a machine to cut a flat square of Aerogel of area 144 cm². If there is a maximum error tolerance in the area of 8 cm², how accurately must the engineer cut on the side, assuming all sides have the same length? How do these numbers relate to $\delta, \, \varepsilon, \, a$ , and $L$ ?

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0.033 cm, $\varepsilon =8, \, \delta =0.33, \, a=12, \, L=144$

25. Use the precise definition of limit to prove that the following limit does not exist: $\underset{x\to 1}{\lim}\dfrac{|x-1|}{x-1}$

26. Using precise definitions of limits, prove that $\underset{x\to 0}{\lim}f(x)$ does not exist, given that $f(x)$ is the ceiling function.

Hint

Try any $\delta <1$ .

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27. Using precise definitions of limits, prove that $\underset{x\to 0}{\lim}f(x)$ does not exist: $f(x)=\begin{cases} 1 & \text{ if } \, x \, \text{is rational} \\ 0 & \text{ if } \, x \, \text{is irrational} \end{cases}$

Hint

28. Using precise definitions of limits, determine $\underset{x\to 0}{\lim}f(x)$ for $f(x)=\begin{cases} x & \text{ if } \, x \, \text{is rational} \\ 0 & \text{ if } \, x \, \text{is irrational} \end{cases}$

Hint

Break into two cases, $x$ rational and $x$ irrational.

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29. Using the function from the previous exercise, use the precise definition of limits to show that $\underset{x\to a}{\lim}f(x)$ does not exist for $a\ne 0$ .

For the following exercises (30-32), suppose that $\underset{x\to a}{\lim}f(x)=L$ and $\underset{x\to a}{\lim}g(x)=M$ both exist. Use the precise definition of limits to prove the following limit laws:

30. $\underset{x\to a}{\lim}(f(x)-g(x))=L-M$

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$f(x)-g(x)=f(x)+(-1)g(x)$

31. $\underset{x\to a}{\lim}[cf(x)]=cL$ for any real constant $c$

Hint

Consider two cases: $c=0$ and $c\ne 0$ .

32. $\underset{x\to a}{\lim}[f(x)g(x)]=LM$ .

Hint

$|f(x)g(x)-LM|=|f(x)g(x)-f(x)M+f(x)M-LM|\le |f(x)||g(x)-M|+|M||f(x)-L|$ .

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Limits: Supplemental Content

Problem Set: The Precise Definition of a Limit

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