12.R: Introduction to Calculus (Review)

12.1: Finding Limits - Numerical and Graphical Approaches

For the exercises 1-6, use the Figure below.

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

Answer

$2$

2) $\lim \limits_{x \to -1^-}f(x)$

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

Answer

does not exist

4) $\lim \limits_{x \to 3}f(x)$

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

Answer

Discontinuous at $x=-1\left (\lim \limits_{x \to a}f(x) \text{ does not exist} \right )$, $x=3\left (\text{ jump discontinuity} \right )$, and $x=7\left (\lim \limits_{x \to a}f(x) \text{ does not exist} \right )$.

6) Using the Table below, estimate $\lim \limits_{x \to 0}f(x)$.

$x$	$F(x)$
−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

Answer

$3$

For the exercises 7-9, 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) $f(x)=\begin{cases} \left | x \right |-1 & \text{ if } x\neq 1 \\ x^3 & \text{ if } x= 1 \end{cases} a=1$

8) $f(x)=\begin{cases} \dfrac{1}{x+1} & \text{ if } x= -2 \\ (x+1)^2 & \text{ if } x\neq -2 \end{cases} a=-2$

Answer

$\lim \limits_{x \to -2}f(x)=1$

9) $f(x)=\begin{cases} \sqrt{x+3} & \text{ if } x<1 \\ -\sqrt[3]{x} & \text{ if } x>1 \end{cases} a=1$

12.2: Finding Limits - Properties of Limits

For the exercises 1-6, find the limits if $\lim \limits_{x \to c} f(x)=-3$ and $\lim \limits_{x \to c} g(x)=5$.

1) $\lim \limits_{x \to c} (f(x)+g(x))$

Answer

$2$

2) $\lim \limits_{x \to c} \dfrac{f(x)}{g(x)}$

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

Answer

$-15$

4) $\lim \limits_{x \to 0^+} f(x), f(x)=\begin{cases} 3x^2+2x+1 & x>0 \\ 5x+3 & x<0 \end{cases}$

5) $\lim \limits_{x \to 0^-} f(x), f(x)=\begin{cases} 3x^2+2x+1 & x>0 \\ 5x+3 & x<0 \end{cases}$

Answer

$3$

6) $\lim \limits_{x \to 3^+} (3x-〚x〛)$

For the exercises 7-11, evaluate the limits using algebraic techniques.

7) $\lim \limits_{h \to 0} \left ( \dfrac{(h+6)^2-36}{h} \right )$

Answer

$12$

8) $\lim \limits_{x \to 25} \left ( \dfrac{x^2-625}{\sqrt{x}-5} \right )$

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

Answer

$-10$

10) $\lim \limits_{x \to 4} \left ( \dfrac{7-\sqrt{12x+1}}{x-4} \right )$

11) $\lim \limits_{x \to 3} \left ( \dfrac{\frac{1}{3}+\frac{1}{x}}{3+x} \right )$

Answer

$-\dfrac{1}{9}$

12.3: Continuity

For the exercises 1-5, use numerical evidence to determine whether the limit exists at $x=a$. If not, describe the behavior of the graph of the function at $x=a$.

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

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

Answer

At $x=4$, the function has a vertical asymptote.

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

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

Answer

removable discontinuity at $a=-\dfrac{5}{2}$

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

For the exercises 6-12, determine where the given function $f(x)$ is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.

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

Answer

continuous on $(-\infty, \infty)$

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

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

Answer

removable discontinuity at $x=2$. $f(2)$ is not defined, but limits exist.

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

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

Answer

discontinuity at $x=0$ and $x=2$. Both $f(0)$ and $f(2)$ are not defined.

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

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

Answer

removable discontinuity at $x=-2$. $f(-2)$ is not defined.

12.4: Derivatives

For the exercises 1-5, find the average rate of change $f(x)=\dfrac{f(x+h)-f(x)}{h}$.

1) $f(x)=3x+2$

2) $f(x)=5$

Answer

$0$

3) $f(x)=\dfrac{1}{x+1}$

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

Answer

$f(x)=\dfrac{\ln (x+h)-\ln (x)}{h}$

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

For the exercises 6-7, find the derivative of the function.

6) $f(x)=4x-6$

Answer

$4$

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

8) Find the equation of the tangent line to the graph of $f(x)$ at the indicated $x$ value. $$f(x)=-x^3+4x;\; x=2 \nonumber$$

Answer

$y=-8x+16$

9) For the following exercise, 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. $$f(x)=\dfrac{x}{\left | x \right |} \nonumber$$

10) Given that the volume of a right circular cone is $V=\dfrac{1}{3}\pi r^2h$ 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 $π$.

Answer

$12\pi$

Practice Test

For the exercises 1-6, use the graph of $f$ in the Figure below.

1) $f(1)$

Answer

$3$

2) $\lim \limits_{x \to -1^+} f(x)$

3) $\lim \limits_{x \to -1^-} f(x)$

Answer

$0$

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

5) $\lim \limits_{x \to -2} f(x)$

Answer

$-1$

6) At what values of $x$ is $f$ discontinuous? What property of continuity is violated?

7) $f(x)=\begin{cases} \dfrac{1}{3}-3 & \text{ if } x\leq 2 \\ x^3+1 & \text{ if } x>2 \end{cases} a=2$

Answer

$\lim \limits_{x \to 2^-} f(x)=-\dfrac{5}{2}a$ and $\lim \limits_{x \to 2^+} f(x)=9$

 Thus, the limit of the function as $x$ approaches $2$ does not exist.

8) $f(x)=\begin{cases} x^3+1 & \text{ if } x<1 \\ 3x^2-1 & \text{ if } x=1\; a=1 \\ -\sqrt{x+3}+4 & \text{ if } x>1 \end{cases}$

For the exercises 9-11, evaluate each limit using algebraic techniques.

9) $\lim \limits_{x \to -5} \left ( \dfrac{\frac{1}{5}+\frac{1}{x}}{10+2x} \right )$

Answer

$-\dfrac{1}{50}$

10) $\lim \limits_{h \to 0} \left ( \dfrac{\sqrt{h^2+25}-5}{h^2} \right )$

11) $\lim \limits_{h \to 0} \left ( \dfrac{1}{h}-\dfrac{1}{h^2+h} \right )$

Answer

$1$

For the exercises 12-13, determine whether or not the given function $f$ is continuous. If it is continuous, show why. If it is not continuous, state which conditions fail.

12) $f(x)=\sqrt{x^2-4}$

13) $f(x)=\dfrac{x^3-4x^2-9x+36}{x^3-3x^2+2x-6}$

Answer

removable discontinuity at $x=3$

For the exercises 14-16, use the definition of a derivative to find the derivative of the given function at $x=a$.

14) $f(x)=\dfrac{3}{5+2x}$

15) $f(x)=\dfrac{3}{\sqrt{x}}$

Answer

$f'(x)=-\dfrac{3}{2a^{\frac{3}{2}}}$

16) $f(x)=2x^2+9x$

17) For the graph in the Figure below, determine where the function is continuous/discontinuous and differentiable/not differentiable.

Answer

discontinuous at $-2,0$, not differentiable at $-2,0, 2$.

For the exercises 18-19, 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.

18) $f(x)=\left | x-2 \right | - \left | x+2 \right |$

19) $f(x)=\dfrac{2}{1+e^{\frac{2}{x}}}$

Answer

not differentiable at $x=0$ (no limit)

For the exercises 20-24, explain the notation in words when the height of a projectile in feet, $s$, is a function of time $t$ in seconds after launch and is given by the function $s(t)$.

20) $s(0)$

21) $s(2)$

Answer

the height of the projectile at $t=2$ seconds

22) $s'(2)$

23) $\dfrac{s(2)-s(1)}{2-1}$

Answer

the average velocity from $t=1$ to $t=2$

24) $s(t)=0$

For the exercises 25-28, use technology to evaluate the limit.

25) $\lim \limits_{x \to 0}\dfrac{\sin (x)}{3x}$

Answer

$\dfrac{1}{3}$

26) $\lim \limits_{x \to 0}\dfrac{\tan ^2(x)}{2x}$

27) $\lim \limits_{x \to 0}\dfrac{\sin (x)(1-\cos (x))}{2x^2}$

Answer

$0$

28) Evaluate the limit by hand.

$$\lim \limits_{x \to 1}f(x), \text{ where } f(x)=\begin{cases} 4x-7 & x\neq 1 \\ x^2-4 & x= 1 \end{cases} \nonumber$$

At what value(s) of $x$ is the function below discontinuous?

$$f(x)=\begin{cases} 4x-7 & x\neq 1 \\ x^2-4 & x= 1 \end{cases} \nonumber$$

For the exercises 29-32, consider the function whose graph appears in Figure.

29) Find the average rate of change of the function from $x=1$ to $x=3$.

Answer

$2$

30) Find all values of $x$ at which $f'(x)=0$.

Answer

$x=1$

31) Find all values of $x$ at which $f'(x)$ does not exist.

32) Find an equation of the tangent line to the graph of $f$ the indicated point: $f(x)=3x^2-2x-6,\; x=-2$

Answer

$y=-14x-18$

For the exercises 33-34, use the function $f(x)=x(1-x)^{\frac{2}{5}}$

33) Graph the function $f(x)=x(1-x)^{\tfrac{2}{5}}$ by entering $f(x)=x\left ((1-x)^2 \right )^{\tfrac{1}{5}}$ and then by entering $f(x)=x\left ((1-x)^{\tfrac{1}{5}} \right )^2$.

34) Explore the behavior of the graph of $f(x)$ around $x=1$ by graphing the function on the following domains, $[0.9, 1.1], [0.99, 1.01], [0.999, 1.001]$, and $[0.9999, 1.0001]$. Use this information to determine whether the function appears to be differentiable at $x=1$.

Answer

The graph is not differentiable at $x=1$ (cusp).

For the exercises 35-42, find the derivative of each of the functions using the definition: $\lim \limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$

35) $f(x)=2x-8$

36) $f(x)=4x^2-7$

Answer

$f'(x)=8x$

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

38) $f(x)=\dfrac{1}{x+2}$

Answer

$f'(x)=-\dfrac{1}{(2+x)^2}$

39) $f(x)=\dfrac{3}{x-1}$

40) $f(x)=-x^3+1$

Answer

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

41) $f(x)=x^2+x^3$

42) $f(x)=\sqrt{x-1}$

Answer

$f'(x)=-\dfrac{1}{2\sqrt{x-1}}$

Contributors and Attributions

• Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus.