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Mathematics LibreTexts

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 lim 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.


2.R: Chapter 2 Review Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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