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# 1.E: Applications of Limits (Exercises)

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## 1.1: An Introduction to Limits

#### Terms and Concepts

1. In your own words, what does it mean to "find the limit of $$f(x)$$ as $$x$$ approaches 3"?
2. An expression of the form $$\frac00$$ is called _____.
3. T/F: The limit of $$f(x)$$ as $$x$$ approaches 5 is $$f(5)$$.
4. Describe three situations where $$\lim\limits_{x\to c}f(x)$$ does not exist.
5. In your own words, what is a difference quotient.

#### Problems

In Exercises 6-16, approximate the given limits both numerically and graphically.

6. $$\lim\limits_{x\to 1}x^2+3x-5$$

7. $$\lim\limits_{x\to 0}x^3-3x^2+x-5$$

8. $$\lim\limits_{x\to 0}\frac{x+1}{x^2+3x}$$

9. $$\lim\limits_{x\to 3}\frac{x^2-2x-3}{x^2-4x+3}$$

10. $$\lim\limits_{x\to-1}\frac{x^2+8x+7}{x^2+6x+5}$$

11. $$\lim\limits_{x\to 2}\frac{x^2+7x+10}{x^2-4x+4}$$

12. $$\lim\limits_{x\to 2}$$, where $$f(x) = \begin{cases}x+2 \quad x\le 2\\ 3x-5 \quad x>2 \end{cases}.$$

13. $$\lim\limits_{x\to 3}$$, where $$f(x) = \begin{cases}x^2-x+1 \quad & x\le 3\\ 2x+1 &x>3 \end{cases}.$$

14. $$\lim\limits_{x\to 0}$$, where $$f(x) = \begin{cases}\cos x \quad & x\le 0\\ x^2+3x+1 &x>0 \end{cases}.$$

15. $$\lim\limits_{x\to \pi/2}$$, where $$f(x) = \begin{cases}\sin x \quad & x\le \pi/2\\ \cos x &x>\pi/2 \end{cases}.$$

In Exercises 16-24, a function $$f$$ and a value $$a$$ are given. Approximate the limit of the difference quotient, $$\lim\limits_{h\to 0}\frac{f(a+h)-f(a)}{h}$$, using $$h=\pm 0.1,\, \pm 0.01.$$

16. $$f(x)=-7x+2,\quad a=3$$

17. $$f(x)=9x+0.06,\quad a=-1$$

18. $$f(x)=x^2+3x-7,\quad a=1$$

19. $$f(x)=\frac{1}{x+1},\quad a=2$$

20. $$f(x)=-4x^2+5x-1,\quad a=-3$$

21. $$f(x)=\ln x,\quad a=5$$

22. $$f(x)=\sin x,\quad a=\pi$$

23. $$f(x)=\cos x,\quad a=\pi$$

## 1.2: Epsilon-Delta Definition of a Limit

#### Terms and Concepts

1. What is wrong with the following "definition" of a limit?

"The limit of $$f(x)$$, as x approaches $$a$$, is $$K''$$ means that given any $$\delta >0$$ there exists $$\epsilon >0$$ such that whenever $$|f(x)-K|<\epsilon$$, we have $$|x-a|<\delta$$.

2. Which is given first in establishing a limit, the x-tolerance or the y-tolerance?

3. T/F: $$\epsilon$$ must always be positive.

4. T/F: $$\delta$$ must always be positive.

#### Problems

In Exercises 5-11, prove the given limit using an $$\epsilon -\delta$$ proof.

5. $$\lim\limits_{x\to5}3-x+-2$$

6. $$\lim\limits_{x\to3}x^2-3=6$$

7. $$\lim\limits_{x\to4}x^2+x-5=15$$

8. $$\lim\limits_{x\to2}x^3-1=7$$

9. $$\lim\limits_{x\to2}5=5$$

10. $$\lim\limits_{x\to0}e^{2x}-1=0$$

11. $$\lim\limits_{x\to0}\sin x = 0$$ (Hint: use the fact that $$|\sin x |\le |x|,$$ with equality only when $$x=0$$.)

## 1.3: Finding Limits Analytically

#### Terms and Concepts

1. Explain in your own words, without using $$ε-δ$$ formality, why $$\lim\limits_{x\to c}b=b$$.

2. Explain in your own words, without using $$ε-δ$$ formality, why $$\lim\limits_{x\to c}x=c$$.

3. What does the text mean when it says that certain functions’ “behavior is ‘nice’ in terms of limits”? What, in particular, is “nice”?

4. Sketch a graph that visually demonstrates the Squeeze Theorem.

5. You are given the following information:

(a) $$\lim\limits_{x\to1}f(x)=0$$

(b)$$\lim\limits_{x\to1}g(x)=0$$

(c)$$\lim\limits_{x\to1}f(x)/g(x) =2$$

What can be said about the relative sizes of $$f(x)$$ and $$g(x)$$ as x approaches 1?

#### Problems

Using:

\begin{align}\lim\limits_{x\to9}f(x)=6 \qquad \lim\limits_{x\to6}f(x)=9 \\ \lim\limits_{x\to9}g(x)=3 \qquad \lim\limits_{x\to6}g(x)=3 \end{align}

evaluate the limits given in Exercises 6-13, where possible. If it is not possible to know, state so.

6. $$\lim\limits_{x\to9}(f(x)+g(x))$$

7. $$\lim\limits_{x\to9}(3f(x)/g(x))$$

8. $$\lim\limits_{x\to9} \left ( \frac{f(x)-2g(x)}{g(x)}\right )$$

9. $$\lim\limits_{x\to6}\left (\frac{f(x)}{3-g(x)}\right )$$

10. $$\lim\limits_{x\to9}g(f(x))$$

11. $$\lim\limits_{x\to6}f(g(x))$$

12. $$\lim\limits_{x\to6}g(f(f(x)))$$

13. $$\lim\limits_{x\to6}f(x)g(x)-f^2(x)+g^2(x)$$

Using

\begin{align}\lim\limits_{x\to1}f(x)=2 \qquad \lim\limits_{x\to10}f(x)=1 \\ \lim\limits_{x\to1}g(x)=0 \qquad \lim\limits_{x\to10}g(x)=\pi \end{align}

evaluate the limits given in Exercises 14-17, where possible. If it is not possible to know, state so.

14. $$\lim\limits_{x\to1}f(x)^{g(x)}$$

15. $$\lim\limits_{x\to10}\cos (g(x))$$

16. $$\lim\limits_{x\to1}f(x)g(x)$$

17. $$\lim\limits_{x\to1}g(5f(x))$$

In Exercises 18-32, evaluate the given limit.

18. $$\lim\limits_{x\to3}x^2-3x+7$$

19. $$\lim\limits_{x\to\pi}\left ( \frac{x-3}{x+5}\right )^7$$

20. $$\lim\limits_{x\to\pi /4}\cos x \sin x$$

21. $$\lim\limits_{x\to 0}\ln x$$

22. $$\lim\limits_{x\to3}4^{{x^3}-8x}$$

23. $$\lim\limits_{x\to\pi/6}\csc x$$

24. $$\lim\limits_{x\to0}\ln (1+x)$$

25. $$\lim\limits_{x\to\pi}\frac{x^2+3x+5}{5x^2-2x-3}$$

26.$$\lim\limits_{x\to\pi}\frac{3x+1}{1-x}$$

27.$$\lim\limits_{x\to6}\frac{x^2-4x-12}{x^2-13x+42}$$

28.$$\lim\limits_{x\to0}\frac{x^2+2x}{x^2-2x}$$

29.$$\lim\limits_{x\to2}\frac{x^2+6x-16}{x^2-3x+2}$$

30.$$\lim\limits_{x\to2}\frac{x^2-5x-14}{x^2+10x+16}$$

31.$$\lim\limits_{x\to-2}\frac{x^2-5x-14}{x^2+10x+16}$$

32.$$\lim\limits_{x\to-1}\frac{x^2+9x+8}{x^2-6x-7}$$\

Use the Squeeze Theorem in Exercises 33-36, where appropriate, to evaluate the given limit.

33. $$\lim\limits_{x\to0} x\sin \left (\frac{1}{x}\right )$$

34. $$\lim\limits_{x\to0}\sin x \cos \left ( \frac{1}{x^2}\right )$$

35. $$\lim\limits_{x\to1}f(x)$$, where $$3x-2\le f(x)\le x^3.$$

36. $$\lim\limits_{x\to3+}f(x),$$ where $$6x-9\le f(x)\le x^2$$ on [0,3].

Exercises 37-40, challenge your understanding of limits but can be evaluated using the knowledge gained in this section.

37. $$\lim\limits_{x\to0}\frac{\sin 3x}{x}$$

38. $$\lim\limits_{x\to0}\frac{\sin 5x}{8x}$$

39. $$\lim\limits_{x\to0}\frac{\ln (1+x)}{x}$$

40. $$\lim\limits_{x\to0}\frac{\sin x}{x}$$, where x is measured in degrees not radians.

## 1.4: One Sided Limits

#### Terms and Concepts

1. What are the three ways in which a limit may fail to exist?

2. T/F: If $$\lim\limits_{x\to1-}f(x)=5$$, then $$\lim\limits_{x\to1}f(x)=5$$

3. T/F: If $$\lim\limits_{x\to1-}f(x)=5$$, then $$\lim\limits_{x\to1+}f(x)=5$$

4. T/F: If $$\lim\limits_{x\to1}f(x)=5$$, then $$\lim\limits_{x\to1-}f(x)=5$$

#### Problems

In Exercises 5-12, evaluate each expression using the given graph of $$f(x)$$.

5.

6.

7.

8.

9.

10.

11.

12.

In Exercises 13-21, evaluate the given limits of the piecewise defined functions $$f$$.

13. $$f(x) = \begin{cases} x+1 \quad &x\le 1\\ x^2-5 &x>1 \end{cases}$$
(a) $$\lim\limits_{x\to1^-}f(x)$$
(b) $$\lim\limits_{x\to0^+}f(x)$$
(c) $$\lim\limits_{x\to1}f(x)$$
(d) $$f(1)$$

14. $$f(x) = \begin{cases} 2x^2+5x-1 \quad &x<0 \\ \sin x &x\ge 0 \end{cases}$$
(a) $$\lim\limits_{x\to0^-}f(x)$$
(b) $$\lim\limits_{x\to0^+}f(x)$$
(c) $$\lim\limits_{x\to0}f(x)$$
(d) $$f(0)$$

15. $$f(x) = \begin{cases} x^-1 \quad &x<-1 \\ x^3+1 &-1\le x \le 1 \\ x^2+1 &x>1 \end{cases}$$
(a) $$\lim\limits_{x\to-1^-}f(x)$$
(b) $$\lim\limits_{x\to1^+}f(x)$$
(c) $$\lim\limits_{x\to-1}f(x)$$
(d) $$f(-1)$$
(e) $$\lim\limits_{x\to1^-}f(x)$$
(f) $$\lim\limits_{x\to1^+}f(x)$$
(g) $$\lim\limits_{x\to1}f(x)$$
(h) $$f(1)$$

16. $$f(x) = \begin{cases} \cos x \quad &x<\pi \\ \sin x &x\ge \pi \end{cases}$$
(a) $$\lim\limits_{x\to\pi^-}f(x)$$
(b) $$\lim\limits_{x\to\pi^+}f(x)$$
(c) $$\lim\limits_{x\to\pi}f(x)$$
(d) $$f(\pi)$$

17. $$f(x) = \begin{cases} 1-\cos ^2 x \quad &x<a \\ \sin^2 x &x\ge a \end{cases}$$, where $$a$$ is a real number.
(a) $$\lim\limits_{x\to a^-}f(x)$$
(b) $$\lim\limits_{x\to a^+}f(x)$$
(c) $$\lim\limits_{x\to a}f(x)$$
(d) $$f(a)$$

18. $$f(x) = \begin{cases} x+1 \quad &x<1 \\ 1 &x=1 \\ x-1 &x>1 \end{cases}$$
(a) $$\lim\limits_{x\to1^-}f(x)$$
(b) $$\lim\limits_{x\to1^+}f(x)$$
(c) $$\lim\limits_{x\to1}f(x)$$
(d) $$f(1)$$

19. $$f(x) = \begin{cases} x^2 \quad &x<2 \\ x+1 &x=2 \\ -x^2+2x+4 &x>2 \end{cases}$$
(a) $$\lim\limits_{x\to2^-}f(x)$$
(b) $$\lim\limits_{x\to2^+}f(x)$$
(c) $$\lim\limits_{x\to2}f(x)$$
(d) $$f(2)$$

20. $$f(x) = \begin{cases} a(x-b)^2+c\quad &x<b \\ a(x-b)+c &x\ge b \end{cases}$$, where a, b and c are real numbers.
(a) $$\lim\limits_{x\to b^-}f(x)$$
(b) $$\lim\limits_{x\to b^+}f(x)$$
(c) $$\lim\limits_{x\to b}f(x)$$
(d) $$f(b)$$

21. $$f(x) = \begin{cases}\frac{|x|}{x} \quad &x\ne 0 \\ 0 &x= 0 \end{cases}$$
(a) $$\lim\limits_{x\to0^-}f(x)$$
(b) $$\lim\limits_{x\to0^+}f(x)$$
(c) $$\lim\limits_{x\to0}f(x)$$
(d) $$f(0)$$

#### Review

22. Evaluate the limit: $$\lim\limits_{x\to -1}\frac{x^2+5x+4}{x^2-3x-4}$$

23. Evaluate the limit: $$\lim\limits_{x\to -4}\frac{x^2-16}{x^2-4x-32}$$

24. Evaluate the limit: $$\lim\limits_{x\to -6}\frac{x^2-15x+54}{x^2-6x}$$

25. Approximate the limit numerically: $$\lim\limits_{x\to 0.4}\frac{x^2-4.4x+1.6}{x^2-0.4x}$$

26. Approximate the limit numerically: $$\lim\limits_{x\to 0.2}\frac{x^2+5.8x-1.2}{x^2-4.2x+0.8}$$

## 1.5: Continuity

#### Terms and Concepts

1. In your own words, describe what it means for a function to be continuous.

2. In your own words, describe what the Intermediate Value Theorem states.

3. What is a “root” of a function?

4. Given functions $$f\text{ and }g$$ on an interval $$I$$, how can the Bisection Method be used to find a value c where $$f(c) = g(c)$$?

5. T/F: If $$f$$ is defined on an open interval containing c, and $$\lim\limits_{x\to c} f(x)$$ exists, then $$f$$ is continuous at c.

6. T/F: If $$f$$ is continuous at c, then $$\lim\limits_{x\to c} f(x)$$ exists

7. T/F: If $$f$$ is continuous at c, then $$\lim\limits_{x\to c^+} f(x)=f(c)$$.

8. T/F: If $$f$$ is continuous on [a, b], then $$\lim\limits_{x\to a^-} f(x)=f(a)$$.

9. T/F: If f is continuous on [0, 1) and [1, 2), then $$f$$ is continuous on [0, 2).

10. T/F: The sum of continuous functions is also continuous.

#### Problems

In Exercises 11-17, a graph of a function $$f$$ is given along with a value $$a$$. Determine if $$f$$ is continuous at $$a$$; if it is not, state why it is not.

11. $$a=1$$

12. $$a=1$$

13. $$a=1$$

14. $$a=0$$

15. $$a=1$$

16. $$a=4$$

17.
(a) $$a=-2$$
(b) $$a=0$$
(c) $$a=2$$

In Exercises 18-21, determine if $$f$$ is continuous at the indicated values. If not, explain why.

18. $$f(x) = \begin{cases} 1 \quad &x=0\\ \frac{\sin x}{x} &x>0 \end{cases}$$
(a) $$x=0$$
(b) $$x=\pi$$

19. $$f(x) = \begin{cases} x^3-x \quad &x<1\\ x-2 &x\ge 1 \end{cases}$$
(a) $$x=0$$
(b) $$x=1$$

20. $$f(x) = \begin{cases} \frac{x^2+5x+4}{x^2 +3x+2} \quad &x\ne -1\\ 3 &x=-1 \end{cases}$$
(a) $$x=-1$$
(b) $$x=10$$

21. $$f(x) = \begin{cases} \frac{x^2-64}{x^2-11x+24} \quad &x\ne 8\\ 5 &x=8 \end{cases}$$
(a) $$x=0$$
(b) $$x=8$$

In Exercises 22-32, give the intervals on which the given function is continuous.

22. $$f(x)=x^2-3x+9$$

23. $$g(x) = \sqrt{x^2-4}$$

24. $$h(k) = \sqrt{1-k}+\sqrt{k+1}$$

25. $$f(t) = \sqrt{5t^2-30}$$

26. $$g(t) = \frac{1}{\sqrt{1-t^2}}$$

27. $$g(x) = \frac{1}{1+x^2}$$

28. $$f(x) = e^x$$

29. $$g(s) = \ln s$$

30. $$h(t) = \cos t$$

31. $$f(k) = \sqrt{1-e^k}$$

32. $$f(x) = \sin (e^x+x^2)$$

33. Let $$f$$ be continuous on [1,5] where $$f(1) = -2 \text{ and }f(5)=-10$$. Does a value $$1<c<5$$ exist such that $$f(c)=-9$$? Why/why not?

34. Let $$g$$ be continuous on [-3,7] where $$g(0)=0 \text{ and }g(2)=25$$. Does a value $$-3<c<7$$ exist such that $$g(c)=15?$$ Why/why not?

35. Let $$f$$ be continuous on [-1,1] where $$f(-1)=-10 \text{ and }f(1)=10$$. Does a value $$-1<c<1$$ exist such that $$f(c)=11?$$ Why/why not?

36. Let $$h$$ be continuous on [-1,1] where $$h(-1)=-10 \text{ and }h(1)=10$$. Does a value $$-1<c<1$$ exist such that $$h(c)=0?$$ Why/why not?

In Exercises 37-40, use the Bisection Method to approximate, accurate to two decimal places, the value of the root of the given function in the given interval.

37. $$f(x) = x^2+2x-4\text{ on }[1,1.5]$$.

38. $$f(x) = \sin x -1/2\text{ on }[0.5,0.55]$$.

39. $$f(x) = e^x-2\text{ on }[0.65,0.7]$$.

40. $$f(x) = \cos x -\sin x \text{ on }[0.7,0.8]$$.

#### Review

41. Let $$f(x) = \begin{cases} x^2-5 \quad &x<5\\ 5x &x\ge 5 \end{cases}$$.
(a) $$\lim\limits_{x\to 5^-}f(x)$$
(b) $$\lim\limits_{x\to 5^+}f(x)$$
(c) $$\lim\limits_{x\to 5}f(x)$$
(d) $$f(5)$$

42. Numerically approximate the following limits:
(a) $$\lim\limits_{x\to 4/5^+}\frac{x^2-8.2x-7.2}{x^2+5.8x+4}$$
(b) $$\lim\limits_{x\to 4/5^-}\frac{x^2-8.2x-7.2}{x^2+5.8x+4}$$

43. Give an example of function $$f(x)$$ for which $$\lim\limits_{x\to 0}f(x)$$ does not exist.

## 1.6: Limits Involving Infinity

#### Terms and Concepts

1. T/F: If $$\lim\limits_{x\to 5}f(x)=\infty$$, then we are implicitly stating that the limit exists.

2. T/F: If $$\lim\limits_{x\to \infty}f(x)=5$$, then we are implicitly stating that the limit exists.

3. T/F: If $$\lim\limits_{x\to 1^-}f(x)=-\infty$$, then $$\lim\limits_{x\to 1^+}f(x)=\infty$$.

4. T/F: If $$\lim\limits_{x\to 5}f(x)=\infty$$, then $$f$$ has a vertical asymptote at $$x=5$$.

5. T/F: $$\infty/0$$ is not an indeterminate form.

6. List 5 indeterminate forms.

7. Construct a function with a vertical asymptote at x = 5 and a horizontal asymptote at y = 5.

8. Let $$\lim\limits_{x\to 7}f(x)=\infty$$. Explain how we know that $$f$$ is/is not continuous at $$x=7$$.

#### Problems

In Exercises 9-14, evaluate the given limits using the graph of the function.

9. $$f(x) = \frac{1}{(x+1)^2}$$
(a) $$\lim\limits_{x\to -1^-}f(x)$$
(b) $$\lim\limits_{x\to -1^+}f(x)$$

10. $$f(x) = \frac{1}{(x-3)(x-5)^2}$$
(a) $$\lim\limits_{x\to 3^-}f(x)$$
(b) $$\lim\limits_{x\to 3^+}f(x)$$
(c) $$\lim\limits_{x\to 3}f(x)$$
(d) $$\lim\limits_{x\to 5^-}f(x)$$
(e) $$\lim\limits_{x\to 5^+}f(x)$$
(f) $$\lim\limits_{x\to 5}f(x)$$

11. $$f(x) = \frac{1}{e^x+1}$$
(a) $$\lim\limits_{x\to -\infty}f(x)$$
(b) $$\lim\limits_{x\to \infty}f(x)$$
(c) $$\lim\limits_{x\to 0^-}f(x)$$
(d) $$\lim\limits_{x\to 0^+}f(x)$$

12. $$f(x) = x^2\sin (\pi x)$$
(a) $$\lim\limits_{x\to -\infty}f(x)$$
(b) $$\lim\limits_{x\to \infty}f(x)$$

13. $$f(x)=\cos (x)$$
(a) $$\lim\limits_{x\to -\infty}f(x)$$
(b) $$\lim\limits_{x\to \infty}f(x)$$

14. $$f(x) = 2^x +10$$
(a) $$\lim\limits_{x\to -\infty}f(x)$$
(b) $$\lim\limits_{x\to \infty}f(x)$$

In Exercises 15-18, numerically approximate the following limits:
(a) $$\lim\limits_{x\to 3^-}f(x)$$
(b) $$\lim\limits_{x\to 3^+}f(x)$$
(c) $$\lim\limits_{x\to 3}f(x)$$

15. $$f(x) = \frac{x^2-1}{x^2-x-6}$$

16. $$f(x) = \frac{x^2+5x-36}{x^3-5x^2+3x+9}$$

17. $$f(x) = \frac{x^2-11x+30}{x^3-4x^2-3x+18}$$

18. $$f(x) = \frac{x^2-9x+18}{x^2-x-6}$$

In Exercises 19-24, identify the horizontal and vertical asymptotes, if any, of the given function.

19. $$f(x) = \frac{2x^2-2x-4}{x^2+x-20}$$

20. $$f(x) = \frac{-3x^2-9x-6}{5x^2-10x-15}$$

21. $$f(x) = \frac{x^2+2-12}{7x^3-14x^2-21x}$$

22. $$f(x) = \frac{x^2-9}{9x-9}$$

23. $$f(x) = \frac{x^2-9}{9x+27}$$

24. $$f(x) = \frac{x^2-1}{-x^2-1}$$

In Exercises 25-28, evaluate the given limit.

25. $$\lim\limits_{x\to \infty}\frac{x^3+2x^2+1}{x-5}$$

26. $$\lim\limits_{x\to \infty}\frac{x^3+2x^2+1}{5-x}$$

27. $$\lim\limits_{x\to \infty}\frac{x^3+2x^2+1}{x^2-5}$$

28. $$\lim\limits_{x\to \infty}\frac{x^3+2x^2+1}{5-x^2}$$

#### Review

29. Use an $$ε − δ$$ proof to show that $$\lim\limits_{x\to 1}5x-2=3$$.

30. Let $$\lim\limits_{x\to 2}f(x)=3\text{ and }\lim\limits_{x\to 2}g(x)=-1$$. Evaluate the following limits.
(a) $$\lim\limits_{x\to 2}(f+g)(x)$$
(b) $$\lim\limits_{x\to 2}(fg)(x)$$
(c) $$\lim\limits_{x\to 2}(f/g)(x)$$
(d) $$\lim\limits_{x\to 2}f(x)^{g(x)}$$

31. Let $$f(x) = \begin{cases}x^2-1 \qquad &x<3 \\ x+5 &x \ge 3 \end{cases}$$. Is $$f$$ continuous everywhere?

32. Evaluate the limit: $$\lim\limits_{x\to c}\ln x$$.