2.7: Chapter 2 Review Exercises
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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)
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.
