Limits

Calculate each limit in Exercises 35–80.

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

Read the section and make your own summary of material.

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) A limit exists if there is some real number that it is equal to.

(b) The limit of $f\left(x\right)$ as $x\to c$ is the value $f\left(c\right)$.

(c) The limit of $f\left(x\right)$ as $x\to c$ might exist even if the value of $f\left(c\right)$ does not.

(d) The two-sided limit of $f\left(x\right)$ as $x\to c$ exists if and only if the left and right limits of $f\left(x\right)$ exists as $x\to c$.

(e) If the graph of $f$ has a vertical asymptote at $x=5$, then $\underset{x\to 5}{\lim }f\left(x\right)=\infty$.

(f) If $\underset{x\to 5}{\lim }f\left(x\right)=\infty$, then the graph of $f$ has a vertical asymptote at $x=5$.

(g) If $\underset{x\to 2}{\lim }f\left(x\right)=\infty$, then the graph of $f$ has a horizontal asymptote at $x=2$.

(h) If $\underset{x\to \infty }{\lim }f\left(x\right)=2$, then the graph of $f$ has a horizontal asymptote at $y=2$.

For each sequence shown, find the next two terms. Then write a general form for the kth term of the sequence.

$$\begin{gathered} 2,6,10,14,18,22,\dots \\ 1,8,27,64,125,216,\dots \\ 3, \frac{3}{4},\frac{3}{9},\frac{3}{16}, \frac{3}{25},\frac{3}{36} , . . . \\ 1,\frac{1}{3},\frac{1}{9},\frac{1}{27},\frac{1}{81},\frac{1}{243},\dots \\ \frac{3}{5},\frac{4}{7},\frac{5}{9},\frac{6}{11},\frac{7}{13},\frac{8}{15},\dots \\ \frac{3}{2},\frac{5}{5},\frac{7}{10},\frac{9}{17},\frac{11}{26},\frac{13}{37},\dots \end{gathered}$$

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading. 

(a) The graph of a function f for which f(2) does not exist but $\underset{x\to 2}{lim} f\left(x\right)$ does exist. 

(b) The graph of a function f for which f(2) exists and $\underset{x\to 2}{lim} f\left(x\right)$ exists, but the two are not equal.

(c) The graph of a function f for which neither f(2) nor $\underset{x\to 2}{lim} f\left(x\right)$ exist. 

construct

construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) The graph of a function f for which $f\left(2\right)$does not exist but $\underset{x\to 2}{\lim }f\left(x\right)$does exist.

(b) The graph of a function f for which $f\left(2\right)$exists and $\underset{x\to 2}{\lim }f\left(x\right)$exists, but the two are not equal.

(c) The graph of a function f for which neither $f\left(2\right)$nor

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

A watermelon dropped from the top of a $50$-foot building has height given by $s\left(t\right)=50-16t^2$ feet and t seconds. Calculate each of the following:

Part (a): The average rate of change of the watermelon over its entire fall, over the first half of its fall, and over the second half of its fall.

Part (b): The average rate of change over the last second, the last half-second, and the last quarter-second of its fall.

If $\underset{x\to 1^{-}}{\lim }f\left(x\right)=5$ and $\underset{x\to 1^{+}}{\lim }f\left(x\right)=5$, what can you say about $\underset{x\to 1}{\lim }f\left(x\right)$? What can you say about $f\left(1\right)$?

If $\underset{x\to 0^{+}}{\lim }f\left(x\right)=-2$, $\underset{x\to 0^{-}}{\lim }f\left(x\right)=3$, and $f\left(0\right)=-2$, what can you say about $\underset{x\to 0}{\lim }f\left(x\right)$?

If $\underset{x\to 2^{+}}{\lim }f\left(x\right)=8$ but $\underset{x\to 2}{\lim }f\left(x\right)$ does not exist, what can you say about $\underset{x\to 2^{-}}{\lim }f\left(x\right)$?

If $\underset{x\to -1^{+}}{\lim }f\left(x\right)=-\infty$ and $\underset{x\to -1^{-}}{\lim }f\left(x\right)=-\infty$, what can you say about $\underset{x\to -1}{\lim }f\left(x\right)$?

If $\underset{x\to -\infty }{\lim }f\left(x\right)=\infty$, $\underset{x\to \infty }{\lim }f\left(x\right)=3$, and $\underset{x\to -1^{+}}{\lim }f\left(x\right)=\infty$, what can you say about any horizontal and vertical asymptotes of $f$?

Consider the sequence $\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},...,\frac{k}{k+1},....$.

(a) What happens to the terms of this sequence as $k$ gets larger and larger? Express your answer in limit notation.

(b) Use a calculator to find a sufficiently larger value of $k$ so that every term past the $kth$ term of this sequence will be within $0.01$ unit of $1$. 

Consider the sequence $\frac{1}{3},\frac{1}{9},\frac{1}{27},\frac{1}{81},\dots ,\frac{1}{3^k},\dots$

(a) What happens to the terms of this sequence as k gets larger and larger? Express your answer in limit notation. 

(b) Find a sufficiently large value of k so that every term past the kth term of this sequence will be less than 0.0001. 

Consider the sequence of sums 1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4, 1 + 2 + 3 + 4 + 5, .... (a) What happens to the terms of this sequence of sums as k gets larger and larger? (b) Find a sufficiently large value of k that will guarantee that every term past the kth term of this sequence of sums is greater than 1000. 

An orange falling from 20 feet has a height of $s\left(t\right)=20-16t^2$ feet when it has fallen for t seconds.

(a) Graph the position function s(t) and find the time that the orange will hit the ground. 

(b) Make a table to record the average rates that the orange is falling during the last second, half-second, quarter-second, and eighth-of-a-second of its fall. 

(c) From the data in your table, make a guess for the instantaneous final velocity of the orange at the moment it hits the ground 

If you are on the moon, then an orange falling from 20 feet has a height of $s\left(t\right)=20-2.65t^2$ feet when it has fallen for t seconds.

(a) Graph the position function s(t) and find the time that the orange will hit the surface of the moon. 

(b) Make a table to record the average rates that the orange is falling during the last second, half-second, quarter-second, and eighth-of-a-second of its fall on the moon. 

(c) From the data in your table, make a guess for the instantaneous final velocity of the orange at the moment it hits the surface of the moon. 

Consider the area between the graph of $f\left(x\right) =\sqrt{ x}$ and the x-axis on [0, 4] .

(a) Use the four rectangles shown on the left to approximate the given area, and then use the eight rectangles shown on the right to obtain another approximation of that area. Be sure to use the fact that the graph shown is that of the function $f\left(x\right) =\sqrt{ x}$ in your calculations.

(b) Describe what would happen if we did similar approximations with more and more rectangles, and make a guess for the resulting limit. 

Consider the area between the graph of $f\left(x\right)=4-x^2$ and the x-axis on [0, 2]. 

(a) Use the four rectangles shown on the left to approximate the given area, and then use the eight rectangles shown on the right to obtain another approximation of that area. Be sure to use the fact that the graph shown is that of the function $f\left(x\right)=4-x^2$ in your calculations. 

(b) Describe what would happen if we did similar approximations with more and more rectangles, and make a guess for the resulting limit. 

Sketch a function that has the following table of values, but whose limit as $x\to \infty$ is equal to $-\infty$:

Sketch a function that has the following table of values, but whose limit as x → 2 does not exist: 

Use a calculator or other graphing utility to graph the function $f\left(x\right)=\frac{x-2}{x^2-x-2}$. 

(a) Show that f(x) is not defined at x = 2. How is this reflected in your calculator graph? 

(b) Use the graph to argue that even though f(2) is undefined, we have $\underset{x\to 2}{lim} f\left(x\right)=\frac{1}{3}$. 

Use a calculator or other graphing utility to graph the function $g\left(x\right)=\frac{x^2-2x+1}{x-1}$. 

(a) Show that g(x) is not defined at x = 1. How is this reflected in your calculator graph? 

(b) Use the graph to argue that even though g(1) is undefined, we have $\underset{x\to 1}{lim} g\left(x\right)=0$. 

Use a calculator or other graphing utility to investigate the graph $f\left(x\right)=x\sin \left(\frac{1}{x}\right)$ near x = 0. Be sure to have your calculator set to radian mode. Use the graphs to make an educated guess for $\underset{x\to 0}{lim} f\left(x\right)$.

Sketch the graphs of functions that have the given limits and values. (There are multiple correct answers.) 

$\underset{x\to -\mathrm{\infty }}{lim} f\left(x\right)=3\text{ and }\underset{x\to \mathrm{\infty }}{lim} f\left(x\right)=-\mathrm{\infty }$

Sketch the graphs of functions that have the given limits and values. (There are multiple correct answers.) 

$\underset{x\to 2}{lim} f\left(x\right)=-4\text{ and }\underset{x\to -\mathrm{\infty }}{lim} f\left(x\right)=-\mathrm{\infty }$

Sketch the graphs of functions that have the given limits and values. (There are multiple correct answers.) 

$\underset{x\to 0^{+}}{lim} f\left(x\right)=\mathrm{\infty }\text{ and }\underset{x\to 0^{-}}{lim} f\left(x\right)=\mathrm{\infty }$

Sketch the graphs of functions that have the given limits and values. (There are multiple correct answers.)  

$\underset{x\to 5^{-}}{lim} f\left(x\right)=3\text{ and }\underset{x\to 5^{+}}{lim} f\left(x\right)=1$

Consider the limit values below:

$\underset{x->-\infty }{\lim }f\left(x\right)=2 and \underset{x->\infty }{\lim }f\left(x\right)=2$

The strategy is to sketch the graph of the function having the above limit values.

Consider the limit values below:

$\underset{x\to 2^{-}}{\lim }f\left(x\right)=\infty ,\underset{x\to 2^{+}}{\lim }f\left(x\right)=-\infty \text{ and }f\left(2\right)=1$

The strategy is to sketch the graph of the function having the above limit values

Consider the limit values below:

$\underset{x\to 3^{-}}{\lim }f\left(x\right)=2,\underset{x\to 3^{+}}{\lim }f\left(x\right)=2\text{ and }f\left(3\right)$

The strategy is to sketch the graph of the function having the above limit values

Consider the limit values below:

$\underset{x\to -\infty }{\lim }f\left(x\right)=-2,\underset{x\to 3}{\lim }f\left(x\right)=\infty \text{ and }f\left(2\right)=-1$

The strategy is to sketch the graph of the function having the above limit values

Consider the limit values below:

$\underset{x\to 2^{-}}{\lim }f\left(x\right)=2,\underset{x\to 2^{+}}{\lim }f\left(x\right)=-1\text{ and }f\left(2\right)=2$

The strategy is to sketch the graph of the function having the above limit values

Consider the limit values below:

$\underset{x\to 2^{-}}{\lim }f\left(x\right)=3,\underset{x\to 2^{+}}{\lim }f\left(x\right)=3\text{ and }f\left(2\right)=0$

The strategy is to sketch the graph of the function having the above limit values

Consider the graph  of the function:

Find the limits of function.

Consider the graph  of the function:

Find the limits of function.

Consider the graph  of the function:

Find the limits of function.

Consider the graph  of the function:

Find the limits of function.

Consider the graph  of the function:

Find the limits of function.

Consider the graph  of the function:

Find the limits of function.

Consider the graph  of the function:

Find the limits of function.

Consider the graph  of the function:

Find the limits of function.

Find the limit of the expression using table

$\underset{x->2^{-}}{\lim }(x^2+x+1)$

Use tables of values to make educated guesses for each of the limits in Exercises 39–52. 

$\underset{x\to 3^{+}}{\lim }\left(1-3x+x^2\right)$

Use tables of values to make educated guesses for each of the limits in Exercises 39–52. 

$\underset{x\to 2}{\lim }\frac{1}{x^2-4}$

Use tables of values to make educated guesses for each of the limits in Exercises 39–52. $\underset{x\to 1}{\lim }\frac{1}{1-x}.$

Use tables of values to make educated guesses for each of the limits in Exercises 39–52. 

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

Use tables of values to make educated guesses for each of the limits in Exercises 39–52. 

$\underset{x\to 5}{\lim }\frac{x-5}{x^2-25}$

Use tables of values to make educated guesses for each of the limits in Exercises 39–52. 

$\underset{x\to 2}{\lim }\frac{3}{4-2^x}$

Use tables of values to make educated guesses for each of the limits in Exercises 39–52. 

$\underset{x\to \infty }{\lim }\left(3e^{-2x}+1\right)$