2.R: Chapter 2 Review Exercises

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True or False. In exercises 1 - 4, justify your answer with a proof or a counterexample.

1) A function has to be continuous at $x=a$ if the $\displaystyle \lim_{x→a}f(x)$ exists.

2) You can use the quotient rule to evaluate $\displaystyle \lim_{x→0}\frac{\sin x}{x}$.

Answer: False, since we cannot have $\displaystyle \lim_{x→0}x=0$ in the denominator.

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 $\displaystyle \lim_{x→a}f(x)$ does not exist, then $f$ is undefined at the point $x=a$.

Answer: False. A jump discontinuity is possible.

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

a. $\displaystyle \lim_{x→−1}f(x)$

b. $\displaystyle \lim_{x→1}f(x)$

c. $\displaystyle \lim_{x→0^+}f(x)$

d. $\displaystyle \lim_{x→2}f(x)$

$A graph of a piecewise function with several segments. The first is a decreasing concave up curve existing for x < -1. It ends at an open circle at (-1, 1). The second is an increasing linear function starting at (-1, -2) and ending at (0,-1). The third is an increasing concave down curve existing from an open circle at (0,0) to an open circle at (1,1). The fourth is a closed circle at (1,-1). The fifth is a line with no slope existing for x > 1, starting at the open circle at (1,1).$

In exercises 6 - 15, evaluate the limit algebraically or explain why the limit does not exist.

6) $\displaystyle \lim_{x→2}\frac{2x^2−3x−2}{x−2}$

Answer: $5$

7) $\displaystyle \lim_{x→0}3x^2−2x+4$

8) $\displaystyle \lim_{x→3}\frac{x^3−2x^2−1}{3x−2}$

Answer: $8/7$

9) $\displaystyle \lim_{x→π/2}\frac{\cot x}{\cos x}$

10) $\displaystyle \lim_{x→−5}\frac{x^2+25}{x+5}$

Answer: DNE

11) $\displaystyle \lim_{x→2}\frac{3x^2−2x−8}{x^2−4}$

12) $\displaystyle \lim_{x→1}\frac{x^2−1}{x^3−1}$

Answer: $2/3$

13) $\displaystyle \lim_{x→1}\frac{x^2−1}{\sqrt{x}−1}$

14) $\displaystyle \lim_{x→4}\frac{4−x}{\sqrt{x}−2}$

Answer: $−4$

15) $\displaystyle \lim_{x→4}\frac{1}{\sqrt{x}−2}$

In exercises 16 - 17, use the squeeze theorem to prove the limit.

16) $\displaystyle \lim_{x→0}x^2\cos(2πx)=0$

Answer: Since $−1≤\cos(2πx)≤1$, then $−x^2≤x^2\cos(2πx)≤x^2$. Since $\displaystyle \lim_{x→0}x^2=0=\lim_{x→0}−x^2$, it follows that $\displaystyle \lim_{x→0}x^2\cos(2πx)=0$.

17) $\displaystyle \lim_{x→0}x^3\sin\left(\frac{π}{x}\right)=0$

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

Answer: $[2,∞]$

In exercises 19 - 20, determine the value of $c$ such that the function remains continuous. Draw your resulting function to ensure it is continuous.

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

20) $f(x)=\begin{cases}\sqrt{x+1}, & \text{if } x>−1\\x^2+c, & \text{if } x≤−1\end{cases}$

In exercises 21 - 22, use the precise definition of limit to prove the limit.

21) $\displaystyle \lim_{x→1}\,(8x+16)=24$

22) $\displaystyle \lim_{x→0}x^3=0$

Answer: $δ=\sqrt[3]{ε}$

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 $t=[0,2]$.

Answer: $0$ m/sec

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