2. Limits

Chapter 2 Review Exercises

True or False. In the following exercises, justify your answer with a proof or a counterexample.

1. A function has to be continuous at x=a if the \underset{x\to a}{\lim}f(x) exists.

2. You can use the quotient rule to evaluate \underset{x\to 0}{\lim}\frac{\sin x}{x}.

Solution

False

3. If there is a vertical asymptote at x=a for the function f(x), then f is undefined at the point x=a.

4. If \underset{x\to a}{\lim}f(x) does not exist, then f is undefined at the point x=a.

Solution

False. A removable discontinuity is possible.

5. Using the graph, find each limit or explain why the limit does not exist.

  1. \underset{x\to -1}{\lim}f(x)
  2. \underset{x\to 1}{\lim}f(x)
  3. \underset{x\to 0^+}{\lim}f(x)
  4. \underset{x\to 2}{\lim}f(x)

image 1, starting at the open circle at (1,1).”>

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

6. \underset{x\to 2}{\lim}\frac{2x^2-3x-2}{x-2}

Solution

5

7. \underset{x\to 0}{\lim}3x^2-2x+4

8. \underset{x\to 3}{\lim}\frac{x^3-2x^2-1}{3x-2}

Solution

8/7

9. \underset{x\to \pi/2}{\lim}\frac{\cot x}{\cos x}

10. \underset{x\to -5}{\lim}\frac{x^2+25}{x+5}

Solution

DNE

11. \underset{x\to 2}{\lim}\frac{3x^2-2x-8}{x^2-4}

12. \underset{x\to 1}{\lim}\frac{x^2-1}{x^3-1}

Solution

2/3

13. \underset{x\to 1}{\lim}\frac{x^2-1}{\sqrt{x}-1}

14. \underset{x\to 4}{\lim}\frac{4-x}{\sqrt{x}-2}

Solution

−4

15. \underset{x\to 4}{\lim}\frac{1}{\sqrt{x}-2}

In the following exercises, use the squeeze theorem to prove the limit.

16. \underset{x\to 0}{\lim}x^2\cos(2\pi x)=0

Solution

Since -1\le \cos (2\pi x)\le 1, then -x^2\le x^2\cos(2\pi x)\le x^2. Since \underset{x\to 0}{\lim}x^2=0=\underset{x\to 0}{\lim}-x^2, it follows that \underset{x\to 0}{\lim}x^2\cos(2\pi x)=0.

17. \underset{x\to 0}{\lim}x^3\sin(\frac{\pi}{x})=0

18. Determine the domain such that the function f(x)=\sqrt{x-2}+xe^x is continuous over its domain.

Solution

[2,\infty)

In the following exercises, determine the value of c such that the function remains continuous. Draw your resulting function to ensure it is continuous.

19. f(x)=\begin{cases} x^2+1 & \text{if} \, x>c \\ 2x & \text{if} \, x \le c \end{cases}

20. f(x)=\begin{cases} \sqrt{x+1} & \text{if} \, x > -1 \\ x^2+c & \text{if} \, x \le -1 \end{cases}

Solution

c=-1

In the following exercises, use the precise definition of limit to prove the limit.

21. \underset{x\to 1}{\lim}(8x+16)=24

22. \underset{x\to 0}{\lim}x^3=0

Solution

\delta =\sqrt[3]{\epsilon}

23. A ball is thrown into the air and the vertical position is given by x(t)=-4.9t^2+25t+5. Use the Intermediate Value Theorem to show that the ball must land on the ground sometime between 5 sec and 6 sec after the throw.

24. A particle moving along a line has a displacement according to the function x(t)=t^2-2t+4, where x is measured in meters and t is measured in seconds. Find the average velocity over the time period t=[0,2].

Solution

0 m/sec

25. From the previous exercises, estimate the instantaneous velocity at t=2 by checking the average velocity within t=0.01 sec.

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