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Mathematics LibreTexts

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.
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6.
146.PNG

7.
147.PNG

8.
148.PNG

9.
149.PNG

10.
1410.PNG

11.
1411.PNG

12.
1412.PNG

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. \(a=1\)
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12. \(a=1\)
1512.PNG

13. \(a=1\)
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14. \(a=0\)
1514.PNG

15. \(a=1\)
1515.PNG

16. \(a=4\)
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17. 
(a) \(a=-2\)
(b) \(a=0\)
(c) \(a=2\)
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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) \(\lim\limits_{x\to -1^+}f(x)\)
169.PNG

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) \(\lim\limits_{x\to 5}f(x)\)
1610.PNG

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) \(\lim\limits_{x\to 0^+}f(x)\)
1611.PNG

12. \(f(x) = x^2\sin (\pi x)\)
(a) \(\lim\limits_{x\to -\infty}f(x)\)
(b) \(\lim\limits_{x\to \infty}f(x)\)
1612.PNG

13. \(f(x)=\cos (x)\)
(a) \(\lim\limits_{x\to -\infty}f(x)\)
(b) \(\lim\limits_{x\to \infty}f(x)\)
1613.PNG

14. \(f(x) = 2^x +10\)
(a) \(\lim\limits_{x\to -\infty}f(x)\)
(b) \(\lim\limits_{x\to \infty}f(x)\)
1614.PNG

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: \(\lim\limits_{x\to c}\ln x\).