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 [latex]x=a[/latex] if the [latex]\underset{x\to a}{\lim}f(x)[/latex] exists.

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

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

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

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

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

A graph of a piecewise function with several segments. The first is a decreasing concave up curve existing for x < -1. It ends at an open circle at (-1, 1). The second is an increasing linear function starting at (-1, -2) and ending at (0,-1). The third is an increasing concave down curve existing from an open circle at (0,0) to an open circle at (1,1). The fourth is a closed circle at (1,-1). The fifth is a line with no slope existing for x > 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. [latex]\underset{x\to 2}{\lim}\frac{2x^2-3x-2}{x-2}[/latex]

7. [latex]\underset{x\to 0}{\lim}3x^2-2x+4[/latex]

8. [latex]\underset{x\to 3}{\lim}\frac{x^3-2x^2-1}{3x-2}[/latex]

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

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

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

12. [latex]\underset{x\to 1}{\lim}\frac{x^2-1}{x^3-1}[/latex]

13. [latex]\underset{x\to 1}{\lim}\frac{x^2-1}{\sqrt{x}-1}[/latex]

14. [latex]\underset{x\to 4}{\lim}\frac{4-x}{\sqrt{x}-2}[/latex]

15. [latex]\underset{x\to 4}{\lim}\frac{1}{\sqrt{x}-2}[/latex]

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

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

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

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

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

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

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

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

21. [latex]\underset{x\to 1}{\lim}(8x+16)=24[/latex]

22. [latex]\underset{x\to 0}{\lim}x^3=0[/latex]

23. A ball is thrown into the air and the vertical position is given by [latex]x(t)=-4.9t^2+25t+5[/latex]. 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 [latex]x(t)=t^2-2t+4[/latex], where [latex]x[/latex] is measured in meters and [latex]t[/latex] is measured in seconds. Find the average velocity over the time period [latex]t=[0,2][/latex].

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