Chapter 12: Introduction to CalculusContent check — Precalculus, Chapter 12: Introduction to Calculus

For the following limit, define $a,f(x),$ and $L.$

Using the graph of the function $y=f\left(x\right)$ shown in Figure 8, estimate the following limits.

ⓐ $\underset{x\to 0^{-}}{\text{lim}}f\left(x\right)$

ⓑ $\underset{x\to 0^{+}}{\text{lim}}f\left(x\right)$

ⓒ $\underset{x\to 0}{lim}f\left(x\right)$

ⓓ $\underset{x\to 2^{-}}{\text{lim}}f\left(x\right)$

ⓔ $\underset{x\to 2^{+}}{\text{lim}}f\left(x\right)$

ⓕ $\underset{x\to 2}{lim}f\left(x\right)$

ⓖ $\underset{x\to 4^{-}}{\text{lim}}f\left(x\right)$

ⓗ $\underset{x\to 4^{+}}{\text{lim}}f\left(x\right)$

ⓘ $\underset{x\to 4}{lim}f\left(x\right)$

Numerically estimate the limit of the following function by making a table:

For the following exercises, estimate the functional values and the limits from the graph of the function $f$ provided in Figure 14.

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

For the following exercises, estimate the functional values and the limits from the graph of the function $f$ provided in Figure 14.

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

For the following exercises, estimate the functional values and the limits from the graph of the function $f$ provided in Figure 14.

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

For the following exercises, estimate the functional values and the limits from the graph of the function $f$ provided in Figure 14.

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

For the following exercises, estimate the functional values and the limits from the graph of the function $f$ provided in Figure 14.

$\underset{x\to 4^{-}}{\lim }f(x)$

For the following exercises, estimate the functional values and the limits from the graph of the function $f$ provided in Figure 14.

$\underset{x\to 4}{\lim }f(x)$

For the following exercises, draw the graph of a function from the functional values and limits provided.

$\underset{x\to 0^{-}}{\lim }f(x)=$ $2$, $\underset{x\to 0^{+}}{\lim }f(x)=$ $–3$, $\underset{x\to 2}{\lim }f(x)=$ $2$, $f(0)=4$, $f(2)=–1$, $f(–3)\text{ does not exist}.$

For the following exercises, draw the graph of a function from the functional values and limits provided.

$\underset{x\to 2^{-}}{\lim }f(x)=$ $0$, $\underset{x\to 2^{+}}{\lim }=–2$, $\underset{x\to 0}{\lim }f(x)=$ $3$, $f(2)=5$, $f(0)$

For the following exercises, draw the graph of a function from the functional values and limits provided.

$\underset{x\to 2^{-}}{\lim }f(x)=$ $2$, $\underset{x\to 2^{+}}{\lim }f(x)=$ $-$ $3$, $\underset{x\to 0}{\lim }f(x)=$ $5$, $f(0)=1$, $f(1)=0$

For the following exercises, draw the graph of a function from the functional values and limits provided.

$\underset{x\to 3^{-}}{\lim }f(x)=$ $0$, $\underset{x\to 3^{+}}{\lim }f(x)=$ $5$, $\underset{x\to 5}{\lim }f(x)=$ $0$, $f(5)=4$, $f(3)\text{ does not exist}.$

For the following exercises, draw the graph of a function from the functional values and limits provided.

$\underset{x\to 4}{\lim }f(x)=$ $6$, $\underset{x\to 6^{+}}{\lim }f(x)=$ $-$ $1$, $\underset{x\to 0}{\lim }f(x)=$ $5$, $f(4)=6$, $f(2)=6$

For the following exercises, draw the graph of a function from the functional values and limits provided.

$\underset{x\to -3}{\lim }f(x)=$ $2$, $\underset{x\to 1^{+}}{\lim }f(x)=$ $-$ $2$, $\underset{x\to 3}{\lim }f(x)=$ $–4$, $f(–3)=0$, $f(0)=0$

For the following exercises, draw the graph of a function from the functional values and limits provided.

$\underset{x\to \pi }{\lim }f(x)=$ ${\pi }^2$, $\underset{x\to –\pi }{\lim }f(x)=$ $\frac{\pi }{2}$, $\underset{x\to 1^{–}}{\lim }f(x)=$ $0$, $f(\pi )=\sqrt{2}$, $f(0)\text{ does not exist}.$

For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as $x$ approaches 0.

$g(x)=$ ${\left(1+x\right)}^{\frac{2}{x}}$

For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as $x$ approaches 0.

$i(x)=$ ${\left(1+x\right)}^{\frac{4}{x}}$

For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as $x$ approaches 0.

Based on the pattern you observed in the exercises above, make a conjecture as to the limit of $f(x)=$ ${\left(1+x\right)}^{\frac{6}{x}},$ $g(x)=$ ${\left(1+x\right)}^{\frac{7}{x}},$ $\text{and }h(x)=$ ${\left(1+x\right)}^{\frac{n}{x}}.$

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

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

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 near $x=a.$ Round answers to two decimal places.

$f(x)=$ $\frac{x^2-x-6}{x^2-9};$ $a=$ $3$

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 near $x=a.$ Round answers to two decimal places.

$f(x)=$ $\frac{x^2–1}{x^2–3x+2};$ $a=$ $1$

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 near $x=a.$ Round answers to two decimal places.

$f(x)=$ $\frac{10-10x^2}{x^2-3x+2};$ $a=$ $1$

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 near $x=a.$ Round answers to two decimal places.

$f(x)=$ $\frac{x}{4x^2+4x+1};$ $a=$ $-$ $\frac{1}{2}$

For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as $x$ approaches the given value.

$\underset{x\to 0}{\lim }\frac{7\tan x}{3x}$

For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as $x$ approaches the given value.

$\underset{x\to 4}{\lim }\frac{x^2}{x-4}$

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

$\underset{x\to 0}{\lim }{e}^{e^{-\frac{1}{x^2}}}$

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

$\underset{x\to -1}{\lim }\frac{\left|x+1\right|}{x+1}$

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

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

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

$\underset{x\to 0}{\lim }\frac{5}{1-e^{\frac{2}{x}}}$

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

According to the Theory of Relativity, the mass $m$ of a particle depends on its velocity $v$. That is

where $m_o$ is the mass when the particle is at rest and $c$ is the speed of light. Find the limit of the mass, $m,$ as $v$ approaches $c^{-}.$

For the following exercises, evaluate the limits algebraically.

$\underset{x\to 2}{\lim }\left(\frac{-5x}{x^2-1}\right)$

For the following exercises, evaluate the limits algebraically.

$\underset{x\to 3}{\lim }\left(\frac{x^2-9}{x-3}\right)$

For the following exercises, evaluate the limits algebraically.

$\underset{x\to \frac{3}{2}}{\lim }\left(\frac{6x^2-17x+12}{2x-3}\right)$

For the following exercises, evaluate the limits algebraically.

$\underset{x\to 3}{\lim }\left(\frac{x^2-9}{x^2-5x+6}\right)$

For the following exercises, evaluate the limits algebraically.

$\underset{x\to 3}{\lim }\left(\frac{x^2+2x-3}{x-3}\right)$

For the following exercises, evaluate the limits algebraically.

$\underset{h\to 0}{\lim }\left(\frac{{\left(2-h\right)}^3-8}{h}\right)$

For the following exercises, evaluate the limits algebraically.

$\underset{h\to 0}{\lim }\left(\frac{\sqrt{5-h}-\sqrt{5}}{h}\right)$

For the following exercises, evaluate the limits algebraically.

$\underset{x\to 9}{\lim }\left(\frac{x^2-81}{3-\sqrt{x}}\right)$

For the following exercises, evaluate the limits algebraically.

$\underset{x\to 0}{\lim }\left(\frac{x}{\sqrt{1+2x}-1}\right)$

For the following exercises, evaluate the limits algebraically.

$\underset{x\to 4}{\lim }\left(\frac{x^3-64}{x^2-16}\right)$

For the following exercises, evaluate the limits algebraically.

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

For the following exercises, evaluate the limits algebraically.

$\underset{x\to 4^{-}}{\lim }\left(\frac{\left|x-4\right|}{4-x}\right)$

For the following exercises, evaluate the limits algebraically.

$\underset{x\to 4}{\lim }\left(\frac{\left|x-4\right|}{4-x}\right)$

For the following exercise, use the given information to evaluate the limits: $\underset{x\to c}{\lim }f(x)=$ $3,$ $\underset{x\to c}{\lim }g\left(x\right)=$ $5$.

$\underset{x\to c}{\lim }\left[2f(x)+\sqrt{g(x)}\right]$

For the following exercise, use the given information to evaluate the limits: $\underset{x\to c}{\lim }f(x)=$ $3,$ $\underset{x\to c}{\lim }g\left(x\right)=$ $5$.

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

For the following exercises, evaluate the following limits.

$\underset{x\to 2}{\lim }\sin \left(\pi x\right)$

For the following exercises, evaluate the following limits.

$$f(x)=\{\begin{array}{ll}2x^2+2x+1,\hfill & x\le 0\hfill \\ x-3,\hfill & x>0\hfill \end{array}\text{; }\underset{x\to 0^{+}}{\lim }f(x)$$

For the following exercises, evaluate the following limits.

$$f(x)=\{\begin{array}{ll}2x^2+2x+1,\hfill & x\le 0\hfill \\ x-3,\hfill & x>0\hfill \end{array}\text{; }\underset{x\to 0}{\lim }f(x)$$

For the following exercises, evaluate the following limits.

$\underset{x\to 2^{+}}{\lim }(2x-[x])$

For the following exercises, evaluate the following limits.

$\underset{x\to 3^{+}}{\lim }\frac{x^2}{x^2-9}$

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

$f(x)=2x^2-1$

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

$f(x)=$ $x^2+$ $4x-$ $100$

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

$f(x)=\cos (x)$

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

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

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

$f(x)=\sqrt{x}$

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

Find an equation that could be represented by Figure 4.

For the following exercises, refer to Figure 5.

What is the left-hand limit of the function as $x$ approaches 0?

For the following exercises, refer to Figure 5.

The height of a projectile is given by $s(t)=$ $-$ $64t^2+$ $192t$ Find the average rate of change of the height from $t=1$ second to $t=1.5$ seconds.

Identify all discontinuities for the following functions as either a jump or a removable discontinuity.

ⓐ $f(x)=$ $\frac{x^2-6x}{x-6}$

ⓑ $$g(x)=\{\begin{array}{ll}\sqrt{x},\hfill & 0\le x<4\hfill \\ 2x,\hfill & x\ge 4\hfill \end{array}$$

For the following exercises, determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$f(x)=$ $\ln |x+$ $3|,$ $a=$ $-$ $3$

For the following exercises, determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$f(x)=$ $\frac{x^2-16}{x+4},$ $a=$ $-$ $4$

For the following exercises, determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$$f\left(x\right)=\{\begin{array}{l}x,x\ne 3\\ 2x,x=3\end{array}a=3$$

For the following exercises, determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$$f\left(x\right)=\{\begin{array}{ll}\frac{1}{2-x},\hfill & x\ne 2\hfill \\ 3,\hfill & x=2\hfill \end{array}a=2$$

For the following exercises, determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$$f\left(x\right)=\{\begin{array}{ll}3+x,\hfill & x<1\hfill \\ x,\hfill & x=1\hfill \\ x^2,\hfill & x>1\hfill \end{array}a=1$$

For the following exercises, determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$$f\left(x\right)=\{\begin{array}{ll}3+2x,\hfill & x<1\hfill \\ x,\hfill & x=1\hfill \\ -x^2,\hfill & x>1\hfill \end{array}a=1$$

For the following exercises, determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$$f\left(x\right)=\{\begin{array}{ll}\frac{x^2-9}{x+3},\hfill & x<-3\hfill \\ x-9,\hfill & x=-3\hfill \\ \frac{1}{x},\hfill & x>-3\hfill \end{array}a=-3$$

For the following exercises, determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

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

For the following exercises, determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$f\left(x\right)=$ $\frac{x^3-9x}{x^2+11x+24},$ $a=$ $-$ $3$

For the following exercises, determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$f(x)=$ $\frac{x}{\left|x\right|},$ $a=$ $0$

For the following exercises, determine whether or not the given function $f$ is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

$f\left(x\right)=$ $x^3-$ $2x-$ $15$

For the following exercises, determine whether or not the given function $f$ is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

$f\left(x\right)=2\cdot 3^{x+4}$

For the following exercises, determine whether or not the given function $f$ is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

$f\left(x\right)=$ $\frac{\left|x-2\right|}{x^2-2x}$

For the following exercises, determine whether or not the given function $f$ is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

$f\left(x\right)=2x+\frac{5}{x}$

For the following exercises, determine whether or not the given function $f$ is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

$f(x)=\ln x^2$

For the following exercises, determine whether or not the given function $f$ is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

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

For the following exercises, determine whether or not the given function $f$ is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

$f\left(x\right)=$ $x^2+$ $\sin \left(x\right)$

For the following exercises, refer to Figure 15. Each square represents one square unit. For each value of $a,$ determine which of the three conditions of continuity are satisfied at $x=a$ and which are not.

$x=-3$

For the following exercises, refer to Figure 15. Each square represents one square unit. For each value of $a,$ determine which of the three conditions of continuity are satisfied at $x=a$ and which are not.

$x=4$

For the following exercises, use a graphing utility to graph the function $f(x)=$ $\sin \left(\frac{12\pi }{x}\right)$ as in Figure 16. Set the x-axis a short distance before and after 0 to illustrate the point of discontinuity.

Evaluate $f(0).$

For the following exercises, use a graphing utility to graph the function $f(x)=$ $\sin \left(\frac{12\pi }{x}\right)$ as in Figure 16. Set the x-axis a short distance before and after 0 to illustrate the point of discontinuity.

What is the domain of $f\left(x\right)?$

For the following exercises, consider the function shown in Figure 17.

What condition of continuity is violated at these points?

For the following exercises, consider the function shown in Figure 17.

Construct a function that passes through the origin with a constant slope of 1, with removable discontinuities at $x=-7$ and $x=1.$

For the following exercises, consider the function shown in Figure 17.

Find the limit $\underset{x\to 1}{\lim }f(x)$ and determine if the following function is continuous at $x=1:$

Using the graph of the function $f(x)=x^3-3x$ shown in Figure 7, estimate: $f(1),$ $f^{\prime }(1),$ $f(0),$ and $f^{\prime }(0).$

A runner runs along a straight east-west road. The function $f\left(t\right)$ gives how many feet eastward of her starting point she is after $t$ seconds. Interpret each of the following as it relates to the runner.

ⓐ $f\left(0\right)=0$

ⓑ $f\left(10\right)=150$

ⓒ $f^{\prime }\left(10\right)=15$

ⓓ $f^{\prime }\left(20\right)=-10$

ⓔ $f\left(40\right)=\mathrm{-100}$

Determine where the function $y=f\left(x\right)$ shown in Figure 18 is continuous and differentiable from the graph.

For the following exercises, use the definition of derivative $\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$ to calculate the derivative of each function.

$f\left(x\right)=-2x+1$

For the following exercises, use the definition of derivative $\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$ to calculate the derivative of each function.

$f\left(x\right)=$ $2x^2+$ $x-$ $3$

For the following exercises, use the definition of derivative $\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$ to calculate the derivative of each function.

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

For the following exercises, use the definition of derivative $\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$ to calculate the derivative of each function.

$f\left(x\right)=$ $\frac{5-2x}{3+2x}$

For the following exercises, use the definition of derivative $\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$ to calculate the derivative of each function.

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

For the following exercises, use the definition of derivative $\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$ to calculate the derivative of each function.

$f(x)=5\pi$

For the following exercises, find the average rate of change between the two points.

$\left(4,\mathrm{-3}\right)$ and $\left(\mathrm{-2},\mathrm{-1}\right)$

For the following exercises, find the average rate of change between the two points.

$\left(7,\mathrm{-2}\right)$ and $\left(7,10\right)$

For the following polynomial functions, find the derivatives.

$f(x)=$ $-$ $3x^2-$ $7x=$ $6$