Jay Abramson

Review Exercises

Finding Limits: A Numerical and Graphical Approach

For the following exercises, use Figure 1.

Graph of a piecewise function with two segments. The first segment goes from (-1, 2), a closed point, to (3, -6), a closed point, and the second segment goes from (3, 5), an open point, to (7, 9), a closed point. — Figure 1

1.

$\lim_{x \to {−1}^{+}} f (x)$

2.

$\lim_{x \to {−1}^{-}} f (x)$

3.

$\lim_{x \to - 1} f (x)$

4.

$\lim_{x \to 3} f (x)$

5.

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

6.

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

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.

$f (x) = {\begin{array}{l} | x | - 1, & i f & x \neq 1 \\ x^{3}, & i f & x = 1 \end{array} a = 1$

8.

$f (x) = {\begin{array}{l} \frac{1}{x + 1}, & i f & x = - 2 \\ {(x + 1)}^{2}, & i f & x \neq - 2 \end{array} a = - 2$

9.

$f (x) = {\begin{array}{l} \sqrt{x + 3}, & i f & x < 1 \\ - \sqrt[3]{x}, & i f & x > 1 \end{array} a = 1$

Finding Limits: Properties of Limits

For the following exercises, find the limits if $\lim_{x \to c} f (x) = −3$ and $\lim_{x \to c} g (x) = 5.$

10.

$\lim_{x \to c} (f (x) + g (x))$

11.

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

12.

$\lim_{x \to c} (f (x) \cdot g (x))$

13.

$\lim_{x \to 0^{+}} f (x), f (x) = {\begin{matrix} 3 x^{2} + 2 x + 1 \\ 5 x + 3 \end{matrix} \begin{matrix} x > 0 \\ x < 0 \end{matrix}$

14.

$\lim_{x \to 0^{-}} f (x), f (x) = {\begin{matrix} 3 x^{2} + 2 x + 1 \\ 5 x + 3 \end{matrix} \begin{matrix} x > 0 \\ x < 0 \end{matrix}$

15.

$\lim_{x \to 3^{+}} (3 x - [x])$

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

16.

$\lim_{h \to 0} (\frac{{(h + 6)}^{2} - 36}{h})$

17.

$\lim_{x \to 25} (\frac{x^{2} - 625}{\sqrt{x} - 5})$

18.

$\lim_{x \to 1} (\frac{- x^{2} - 9 x}{x})$

19.

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

20.

$\lim_{x \to - 3} (\frac{\frac{1}{3} + \frac{1}{x}}{3 + x})$

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.

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

23.

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

24.

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

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.

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

27.

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

28.

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

29.

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

30.

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

31.

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

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.

$f (x) = 3 x + 2$

34.

$f (x) = 5$

35.

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

36.

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

37.

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

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

38.

$f (x) = 4 x - 6$

39.

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

40.

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

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

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.

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

42.

Given that the volume of a right circular cone is $V = \frac{1}{3} π r^{2} h$ 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 $π$