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 of $f(x)$, find each of the following or explain why it does not exist.

1. $\underset{x\to -1^-}{\lim}f(x)$
2. $\underset{x\to -1^+}{\lim}f(x)$
3. $\underset{x\to -1}{\lim}f(x)$
4. $\underset{x\to 1}{\lim}f(x)$
5. $f(1)$
6. $\underset{x\to 0^+}{\lim}f(x)$
7. $\underset{x\to 0^-}{\lim}f(x)$
8. $\underset{x\to 0}{\lim}f(x)$
9. $\underset{x\to 2}{\lim}f(x)$

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

16. $\underset{x\to 2^-}{\lim}\frac{\frac{1}{2}-\frac{1}{x}}{(x-2)^2}$

Solution

$−\infty$

17. $\underset{x\to 1^-}{\lim}\frac{|x-1|}{x^2-1}$

18. $\underset{t\to 5}{\lim}\frac{t-5}{\sqrt{t-4}-1}$

Solution

2

19. $\underset{x\to2}{\lim}\frac{x^2+x-6}{4-x^2}$

20. $\underset{x\to 4}{\lim}\frac{\sqrt{8-x}-2}{4-x}$

Solution

$\frac{1}{4}$

21. $\underset{x\to 0}{\lim}\frac{\frac{1}{x+7}-\frac{1}{7}}{x}$

22. $\underset{x\to -4^-}{\lim}\frac{\frac{1}{4} + \frac{1}{x}}{x+4}$

Solution

$-\frac{1}{16}$

23. $\underset{x\to 7^-}{\lim}\frac{-1}{(x-7)^{2021}}$

24. $\underset{x\to -6^-}{\lim}\frac{2x+12}{|x+6|}$

Solution

$-2$

25. $\underset{x\to -6}{\lim}\frac{2x+12}{|x+6|}$

In the following exercises, evaluate the limits to infinity.

26. $\underset{x\to -\infty}{\lim}\frac{\sqrt{11x^2+4x}}{5-4x}$

Solution

$\frac{\sqrt{11}}{4}$

27. $\underset{x\to -\infty}{\lim}\frac{\sqrt{7x^2-4x}}{2x-3}$

28. $\underset{x\to \infty}{\lim}(x-\sqrt{x+1})$

Solution

$\infty$

29. $\underset{x\to \infty}{\lim}\sqrt{\frac{3x^2-1}{x+9x^2}}$

30. $\underset{x\to \infty}{\lim}\sqrt{\frac{4x^2-1}{x+3x^2}}$

Solution

$\frac{2}{\sqrt{3}}$

31. $\underset{x\to -\infty}{\lim}\frac{3-x^2}{\sqrt[4]{x^8-4}}$

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

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

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

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

34. $\underset{x\to \infty }{\text{lim}} \frac{1}{x} \cos (2x)$

Solution

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

In the following exercises, determine the value of $c$ such that the function is continuous for the given value of $x$.

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

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

Solution

$c=-1$

37. $f(x)=\begin{cases} \frac{c^2}{3} (x+3) & \text{if} \, x < 0 \\ c+2 & \text{if} \, x = 0 \\ \frac{2}{3}(cx)^2 +1 & \text{if} \, x > 0 \end{cases}$

38. $f(x)=\begin{cases} x+c & \text{if} \, x \le 3 \\ \frac{1}{x} & \text{if} \, x > 3\end{cases}$

Solution

$c=-\frac{8}{3}$

In the following exercises, determine all horizontal and vertical asymptotes.

39. $f(x) = \frac{3x^2-1}{2x^2+7x-4}$

40. $f(x) = \frac{4x^3 -2x}{x^3-1}$

Solution

Horizontal: $y=4$, Vertical: $x = 1$

41. $f(x) = \frac{\sqrt{2x^2+3}}{x^2-2x-3}$

42. $f(x) = \frac{2x^2-1}{\sqrt{5x^4+2}}$

Solution

Horizontal: $y=\frac{2}{\sqrt{5}}$, Vertical: none

In the following exercises, use the Intermediate Value Theorem to show that the given functions have an x-intercept in the given interval.

43. $x^8 - 4x^3 = x + 1$ on the interval $[-1,1]$

44. $f(x) =\frac{3}{x^4} -x^2+2$ on the interval $[-2,-1]$

Solution

Since $f(x)$ is continuous on [-2,-1] and $f(-2) < 0$ and $f(-1) > 0$, then by IVT, there exists a root on the given interval.

45. 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.

46. 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

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

47. $\underset{x\to 1}{\lim}(8x+16)=24$

48. $\underset{x\to 0}{\lim}x^3=0$

Solution

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