12.7.1: Review Exercises

Review Exercises

Finding Limits: A Numerical and Graphical Approach

For the following exercises, use Figure 1.

Figure 1

1.

$\underset{x\to {\mathrm{-1}}^{+}}{\lim }f(x)$

2.

$\underset{x\to {\mathrm{-1}}^{-}}{\lim }f(x)$

3.

$\underset{x\to -1}{\lim }f(x)$

4.

$\underset{x\to 3}{\lim }f(x)$

5.

At what values of $x$ is the function discontinuous? What condition of continuity is violated?

6.

Using Table 1, estimate $\underset{x\to 0}{\lim }f(x).$

$x$	$F(x)$
−0.1	2.875
−0.01	2.92
−0.001	2.998
0	Undefined
0.001	2.9987
0.01	2.865
0.1	2.78145
0.15	2.678

Table 1

For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as $x$ approaches $a.$ If the function has limit as $x$ approaches $a,$ state it. If not, discuss why there is no limit.

7.

$f(x)=\{\begin{array}{lll}\left|x\right|-1,\hfill & if\hfill & x\ne 1\hfill \\ x^3,\hfill & if\hfill & x=1\hfill \end{array}a=1$

8.

$f(x)=\{\begin{array}{lll}\frac{1}{x+1},\hfill & if\hfill & x=-2\hfill \\ {(x+1)}^2,\hfill & if\hfill & x\ne -2\hfill \end{array}a=-2$

9.

Finding Limits: Properties of Limits

For the following exercises, find the limits if $\underset{x\to c}{\lim }f\left(x\right)=\mathrm{-3}$ and $\underset{x\to c}{\lim }g\left(x\right)=5.$

10.

$\underset{x\to c}{\lim }\left(f(x)+g(x)\right)$

11.

$\underset{x\to c}{\lim }\frac{f(x)}{g(x)}$

12.

$\underset{x\to c}{\lim }\left(f(x)\cdot g(x)\right)$

13.

14.

15.

$\underset{x\to 3^{+}}{\lim }\left(3x-[x]\right)$

For the following exercises, evaluate the limits using algebraic techniques.

16.

$\underset{h\to 0}{\lim }\left(\frac{{\left(h+6\right)}^2-36}{h}\right)$

17.

$\underset{x\to 25}{\lim }\left(\frac{x^2-625}{\sqrt{x}-5}\right)$

18.

$\underset{x\to 1}{\lim }\left(\frac{-x^2-9x}{x}\right)$

19.

$\begin{array}{c}\lim \\ {}^{x\to 4}\end{array}\frac{7-\sqrt{12x+1}}{x-4}$

20.

$\underset{x\to -3}{\lim }\left(\frac{\frac{1}{3}+\frac{1}{x}}{3+x}\right)$

Continuity

For the following exercises, use numerical evidence to determine whether the limit exists at $x=a.$ If not, describe the behavior of the graph of the function at $x=a.$

21.

$f(x)=\frac{-2}{x-4};a=4$

22.

$f(x)=\frac{-2}{{\left(x-4\right)}^2};a=4$

23.

$f(x)=\frac{-x}{x^2-x-6};a=3$

24.

$f(x)=\frac{6x^2+23x+20}{4x^2-25};a=-\frac{5}{2}$

25.

$f(x)=\frac{\sqrt{x}-3}{9-x};a=9$

For the following exercises, determine where the given function $f(x)$ is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.

26.

$f(x)=x^2-2x-15$

27.

$f(x)=\frac{x^2-2x-15}{x-5}$

28.

$f(x)=\frac{x^2-2x}{x^2-4x+4}$

29.

$f(x)=\frac{x^3-125}{2x^2-12x+10}$

30.

$f(x)=\frac{x^2-\frac{1}{x}}{2-x}$

31.

$f(x)=\frac{x+2}{x^2-3x-10}$

32.

$f(x)=\frac{x+2}{x^3+8}$

Derivatives

For the following exercises, find the average rate of change $\frac{f(x+h)-f(x)}{h}.$

33.

$f(x)=3x+2$

34.

$f(x)=5$

35.

$f(x)=\frac{1}{x+1}$

36.

$f(x)=\ln (x)$

37.

$f(x)=e^{2x}$

For the following exercises, find the derivative of the function.

38.

$f(x)=4x-6$

39.

$f(x)=5x^2-3x$

40.

Find the equation of the tangent line to the graph of $f\left(x\right)$ at the indicated $x$ value.

$f(x)=-x^3+4x$; $x=2.$

For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.

41.

$f(x)=\frac{x}{\left|x\right|}$

42.

Given that the volume of a right circular cone is $V=\frac{1}{3}\pi r^{2}h$ and that a given cone has a fixed height of 9 cm and variable radius length, find the instantaneous rate of change of volume with respect to radius length when the radius is 2 cm. Give an exact answer in terms of $\pi$