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## Educators

AK
+ 63 more educators

### Problem 1

Explain in your own words what is meant by the equation $\displaystyle \lim_{x\to 2} f(x) = 5$
Is it possible for this statement to be true and yet $f(2) = 3$?
Explain.

AK
Anjali K.

### Problem 2

Explain what it means to say that
$\displaystyle \lim_{x\to1^-}f(x) = 3$ and $\displaystyle \lim_{x\to1^+}f(x) = 7$

In this situation is it possible that $\displaystyle \lim_{x\to1}f(x)$ exists?
Explain.

Dakarai H.

### Problem 3

Explain the meaning of each of the following.
(a) $\displaystyle \lim_{x\to-3}f(x) = \infty$ (b) $\displaystyle \lim_{x\to4^+}f(x) = - \infty$

Ma. Theresa A.

### Problem 4

Use the given graph of $f$ to state the value of each quantity, if it exists. If it does not exist, explain why.

(a) $\displaystyle \lim_{x\to 2^-}f(x)$
(b) $\displaystyle \lim_{x\to 2^+}f(x)$
(c) $\displaystyle \lim_{x\to 2}f(x)$
(d) $f(2)$
(e) $\displaystyle \lim_{x\to 4}f(x)$
(f) $f(4)$

Aparna S.

### Problem 5

For the function $f$ whose graph is given, state the value of each quantity, if it exists. If it does not, explain why.

(a) $\displaystyle \lim_{x\to 1}f(x)$
(b) $\displaystyle \lim_{x\to 3^-}f(x)$
(c) $\displaystyle \lim_{x\to 3^+}f(x)$
(d) $\displaystyle \lim_{x\to 3}f(x)$
(e) $f(3)$

DM
David M.

### Problem 6

For the function $h$ whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why.

(a) $\displaystyle \lim_{x\to -3^-}h(x)$
(b) $\displaystyle \lim_{x\to -3^+}h(x)$
(c) $\displaystyle \lim_{x\to -3}h(x)$
(d) $h(-3)$
(e) $\displaystyle \lim_{x\to 0^-}h(x)$
(f) $\displaystyle \lim_{x\to 0^+}h(x)$
(g) $\displaystyle \lim_{x\to 0}h(x)$
(h) $h(0)$
(i) $\displaystyle \lim_{x\to 2}h(x)$
(j) $h(2)$
(k) $\displaystyle \lim_{x\to 5^+}h(x)$
(l) $\displaystyle \lim_{x\to 5^-}h(x)$

DM
David M.

### Problem 7

For the function $g$ whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why.

(a) $\displaystyle \lim_{t\to 0^-}g(t)$
(b) $\displaystyle \lim_{t\to 0^+}g(t)$
(c) $\displaystyle \lim_{t\to 0}g(t)$
(d) $\displaystyle \lim_{t\to 2^-}g(t)$
(e) $\displaystyle \lim_{t\to 2^+}g(t)$
(f) $\displaystyle \lim_{t\to 2}g(t)$
(g) $g(2)$
(h) $\displaystyle \lim_{t\to 4}g(t)$

DM
David M.

### Problem 8

For the function $A$ whose graph is shown, state the following.

(a) $\displaystyle \lim_{x\to -3}A(x)$
(b) $\displaystyle \lim_{x\to 2^-}A(x)$
(c) $\displaystyle \lim_{x\to 2^+}A(x)$
(d) $\displaystyle \lim_{x\to -1}A(x)$
(e) The equations of the vertical asymptotes

DM
David M.

### Problem 9

For the function $f$ whose graph is shown, state the following.

(a) $\displaystyle \lim_{x\to -7}f(x)$
(b) $\displaystyle \lim_{x\to -3}f(x)$
(c) $\displaystyle \lim_{x\to 0}f(x)$
(d) $\displaystyle \lim_{x\to 6^-}f(x)$
(e) $\displaystyle \lim_{x\to 6^+}f(x)$
(f) The equations of the vertical asymptotes.

Aparna S.

### Problem 10

A patient receives a 150-mg injection of a drug every 4 hours. The graph shows the amount $f(t)$ of the drug in the bloodstream after $t$ hours. Find
$\displaystyle \lim_{t\to 12^-}f(t)$ and $\displaystyle \lim_{t\to 12^+}f(t)$
and explain the significance of these one-sided limits.

DM
David M.

### Problem 11

Sketch the graph of the function and use it to determine the values of $a$ for which $\displaystyle \lim_{x\to a}f(x)$ exists.
$$f(x) = \left\{ \begin{array}{ll} 1 + x & \mbox {if  x < -1 }\\ x^2 & \mbox{if  -1 \le x < 1}\\ 2 - x & \mbox{if  x \ge 1 } \end{array} \right.$$

Leon D.

### Problem 12

Sketch the graph of the function and use it to determine the values of $a$ for which $\displaystyle \lim_{x\to a}f(x)$ exists.
$f(x) = \left\{ \begin{array}{ll} 1 + \sin x & \mbox{if$ x < 0 $}\\ \cos x & \mbox{if$ 0 \le x \le \pi $}\\ \sin x & \mbox{if$ x > \pi $} \end{array} \right.$

Leon D.

### Problem 13

Use the graph of the function $f$ to state the value of each limit, if it exists. If it does not exist, explain why.
(a) $\displaystyle \lim_{x \to 0^-}f(x)$
(b) $\displaystyle \lim_{x \to 0^+}f(x)$
(c) $\displaystyle \lim_{x \to 0}f(x)$

$\displaystyle f(x) = \frac{1}{1+e^{1/x}}$

DM
David M.

### Problem 14

Use the graph of the function $f$ to state the value of each limit, if it exists. If it does not exist, explain why.
(a) $\displaystyle \lim_{x \to 0^-}f(x)$
(b) $\displaystyle \lim_{x \to 0^+}f(x)$
(c) $\displaystyle \lim_{x \to 0}f(x)$

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

Daniel J.

### Problem 15

Sketch the graph of an example of a function $f$ that satisfies all of the given conditions.

$\displaystyle \lim_{x \to 0^-}f(x) = -1$, $\displaystyle \lim_{x \to 0^+}f(x) = 2$, $f(0) = 1$

Stark L.

### Problem 16

Sketch the graph of an example of a function $f$ that satisfies all of the given conditions.

$\displaystyle \lim_{x \to 0}f(x) = 1$, $\displaystyle \lim_{x \to 3^-}f(x) = -2$, $\displaystyle \lim_{x \to 3^+}f(x) = 2$, $f(0) = -1$, $f(3) = 1$

Daniel J.

### Problem 17

Sketch the graph of an example of a function $f$ that satisfies all of the given conditions.

$\displaystyle \lim_{x \to 3^+}f(x) = 4$, $\displaystyle \lim_{x \to 3^-}f(x) = 2$,
$\displaystyle \lim_{x \to -2}f(x) = 2$, $f(3) = 3$, $f(-2) = 1$

Leon D.

### Problem 18

Sketch the graph of an example of a function $f$ that satisfies all of the given conditions.

$\displaystyle \lim_{x \to 0^-}f(x) = 2$, $\displaystyle \lim_{x \to 0^+}f(x) = 0$,
$\displaystyle \lim_{x \to 4^-}f(x) = 3$, $\displaystyle \lim_{x \to 4^+}f(x) = 0$, $f(0) = 2$, $f(4) = 1$

IG
Ian G.

### Problem 19

Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).

$\displaystyle \lim_{x \to 3}\frac{x^2 - 3x}{x^2 - 9}$,
$x$ = 3.1, 3.05, 3.01, 3.001, 3.0001, 2.9, 2.95, 2.99, 2.999, 2.9999

DM
David M.

### Problem 20

Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).

$\displaystyle \lim_{x \to -3}\frac{x^2 - 3x}{x^2 - 9}$,
$x$ = -2.5, -2.9, -2.95, -2.99, -2.999, -2.9999, -3.5, -3.1, -3.05, -3.01, -3.001, -3.0001

Carolyn B.

### Problem 21

Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).

$\displaystyle \lim_{t \to 0}\frac{e^{5t} - 1}{t}$, $t = \pm 0.5, \pm 0.1, \pm 0.01, \pm 0.001, \pm 0.0001$

Paul C.

### Problem 22

Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).

$\displaystyle \lim_{h \to 0}\frac{(2+h)^5 - 32}{h}$,
$h = \pm 0.5, \pm 0.1, \pm 0.01, \pm 0.001, \pm 0.0001$

Oswaldo J.

### Problem 23

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically.

$\displaystyle \lim_{x \to 4}\frac{\ln x - \ln 4}{x-4}$

AK
Anjali K.

### Problem 24

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically.

$\displaystyle \lim_{p \to -1}\frac{1+p^9}{1+p^{15}}$

Daniel J.

### Problem 25

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically.

$\displaystyle \lim_{\theta \to 0}\frac{\sin 3\theta}{\tan 2\theta}$

Satyam G.

### Problem 26

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically.

$\displaystyle \lim_{t \to 0}\frac{5^t - 1}{t}$

Ma. Theresa A.

### Problem 27

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically.

$\displaystyle \lim_{x \to 0^+}x^x$

Leon D.

### Problem 28

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically.

$\displaystyle \lim_{x \to 0^+}x^2 \ln x$

Daniel J.

### Problem 29

(a) By graphing the function $f(x) = (\cos 2x - \cos x)/x^2$ and zooming in toward the point where the graph crosses the $y$ -axis , estimate the value of $\displaystyle \lim_{x \to 0}f(x)$.

(b) Check your answer in part (a) by evaluating $f(x)$ for values of $x$ that approach 0.

Daniel J.

### Problem 30

(a) Estimate the value of $$\lim_{x \to 0}\frac{\sin x}{\sin \pi x}$$
by graphing the function $f(x) = (\sin x)/(\sin \pi x)$. State your answer correct to two decimal places.

(b) Check your answer in part (a) by evaluating $f(x)$ for values of $x$ that approach 0.

Daniel J.

### Problem 31

Determine the infinite limit.

$\displaystyle \lim_{x \to 5^+}\frac{x+1}{x-5}$

DM
David M.

### Problem 32

Determine the infinite limit.

$\displaystyle \lim_{x \to 5^-}\frac{x+1}{x-5}$

Suman Saurav T.

### Problem 33

Determine the infinite limit.

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

tj
Tanvi J.

### Problem 34

Determine the infinite limit.

$\displaystyle \lim_{x \to 3^-}\frac{\sqrt{x}}{(x-3)^5}$

Ma. Theresa A.

### Problem 35

Determine the infinite limit.

$\displaystyle \lim_{x \to 3^+}\ln (x^2 - 9)$

Suman Saurav T.

### Problem 36

Determine the infinite limit.

$\displaystyle \lim_{x \to 0^+}\ln (\sin x)$

Ma. Theresa A.

### Problem 37

Determine the infinite limit.

$\displaystyle \lim_{x \to (\pi/2)^+}\frac{1}{x}\sec x$

DM
David M.

### Problem 38

Determine the infinite limit.

$\displaystyle \lim_{x \to \pi^-}\cot x$

Ma. Theresa A.

### Problem 39

Determine the infinite limit.

$\displaystyle \lim_{x \to 2\pi^-}x\csc x$

Ma. Theresa A.

### Problem 40

Determine the infinite limit.

$\displaystyle \lim_{x \to 2^-}\frac{x^2 - 2x}{x^2 - 4x + 4}$

Ma. Theresa A.

### Problem 41

Determine the infinite limit.

$\displaystyle \lim_{x \to 2^+}\frac{x^2 - 2x -8}{x^2 -5x +6}$

Leon D.

### Problem 42

Determine the infinite limit.

$\displaystyle \lim_{x \to 0^+}\left( \frac{1}{x} - \ln x \right)$

Ma. Theresa A.

### Problem 43

Determine the infinite limit.

$\displaystyle \lim_{x \to 0}(\ln x^2 - x^{-2})$

Ma. Theresa A.

### Problem 44

(a) Find the vertical asymptotes of the function
$$y = \frac{x^2 +1}{3x - 2x^2}$$

Leon D.

### Problem 45

Determine $\displaystyle \lim_{x \to 1^-}\frac{1}{x^3 - 1}$ and $\displaystyle \lim_{x \to 1^+}\frac{1}{x^3 - 1}$

(a) by evaluating $f(x) = 1/(x^3 - 1)$ for values of $x$ that approach 1 from the left and from the right,
(b) by reasoning as in Example 9, and
(c) from a graph of $f$.

Daniel J.

### Problem 46

(a) By graphing the function $f(x) = (\tan 4x)/x$ and zooming in toward the point where the graph crosses the $y$ -axis, estimate the value of $\displaystyle \lim_{x \to 0}f(x)$.

(b) Check your answer in part (a) by evaluating $f(x)$ for values of $x$ that approach 0.

Daniel J.

### Problem 47

(a) Estimate the value of the limit $\displaystyle \lim_{x \to 0}(1 + x)^{1/x}$ to five decimal places. Does this number look familiar?

(b) Illustrate part (a) by graphing the function $y = (1 + x)^{1/x}$.

Daniel J.

### Problem 48

(a) Graph the function $f(x) = e^x + \ln | x - 4 |$ for $0 \le x \le 5$. Do you think the graph is an accurate representation of $f$?

(b) How would you get a graph that represents $f$ better?

Daniel J.

### Problem 49

(a) Evaluate the function $f(x) = x^2 - (2^x/1000)$ for $x$ = 1, 0.8, 0.6, 0.4, 0.2, 0.1, and 0.05, and guess the value of $$\lim_{x \to 0}\left( x^2 - \frac{2^x}{1000} \right)$$

(b) Evaluate $f(x)$ for $x$ = 0.04, 0.02, 0.01, 0.005, 0.003, and 0.001. Guess again.

Daniel J.

### Problem 50

(a) Evaluate $h(x) = (\tan x - x)/x^3$ for $x$ = 1, 0.5, 0.1, 0.05, 0.01, and 0.005.

(b) Guess the value of $\displaystyle \lim_{x \to 0}\frac{\tan x - x}{x^3}$.

(c) Evaluate $h(x)$ for successively smaller values of $x$ until you finally reach a value of 0 for $h(x)$. Are you still confident that your guess in part (b) is correct? Explain why you eventually obtained 0 values. (In Section 4.4 a method of evaluating this limit will be explained.)

(d) Graph the function $h$ in the viewing rectangle $[-1, 1]$ by $[0, 1]$. Then zoom in toward the point where the graph crosses the $y$ -axis to estimate the limit of $h(x)$ as $x$ approaches 0. Continue to zoom in until you observe distortions in the graph of $h$. Compare with the results of part (c).

Oswaldo J.

### Problem 51

Graph the function $f(x) = \sin (\pi/x)$ of Example 4 in the viewing rectangle $[-1, 1]$ by $[-1, 1]$. Then zoom in toward the origin several times. Comment on the behavior of this function.

Leon D.

### Problem 52

Consider the function $f(x) = \tan \frac{1}{x}$.

(a) Show that $f(x) = 0$ for $x = \frac{1}{\pi}, \frac{1}{2\pi}, \frac{1}{3\pi}, ...$

(b) Show that $f(x) = 1$ for $x = \frac{4}{\pi}, \frac{4}{5\pi}, \frac{4}{9\pi}, ...$

(c) What can you conclude about $\displaystyle \lim_{x \to 0^+}\tan \frac{1}{x}$?

Daniel J.

### Problem 53

Use a graph to estimate the equations of all the vertical asymptotes of the curve
$y = \tan (2 \sin x)$ $-\pi \le x \le \pi$
Then find the exact equations of these asymptotes.

Daniel J.

### Problem 54

In the theory of relativity, the mass of a particle with velocity $v$ is
$$m = \frac{m_0}{\sqrt{1 - v^2/c^2}}$$
where $m_0$ is the mass of the particle at rest and $c$ is the speed of light. What happens as $v \to c^-$?

Linda H.
$$\lim_{x \to 1}\frac{x^3 - 1}{\sqrt{x} - 1}$$
(b) How close to 1 does $x$ have to be to ensure that the function in part (a) is within a distance 0.5 of its limit?