Limits of a Function

Calculus 1 / Ab

662 Practice Problems

04:08
Calculus of a Single Variable

In Exercises 63-66, use the definition of limits at infinity to prove the limit.
$$\lim _{x \rightarrow \infty} \frac{2}{\sqrt{x}}=0$$

Applications of Differentiation
Limits at Infinity
Carlos P.
03:56
Essential Calculus Early Transcendentals

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) $\lim _{t \rightarrow 0^{-}} g(t) \quad$ (b) $\lim _{t \rightarrow 0^{+}} g(t)$ (c) $\lim _{t \rightarrow 0} g(t)$
(d) $\lim _{t \rightarrow 2^{-}} g(t) \quad$ (e) $\lim _{t \rightarrow 2^{+}} g(t)$ $\lim _{t \rightarrow 2} g(t)$
$(\mathrm{g}) g(2) \quad$ (h) $\lim _{t \rightarrow 4} g(t)$

FUNCTIONS AND LIMITS
The Limit of a Function
Wesley H.
03:35
Essential Calculus Early Transcendentals

Use the given graph of $f$ to state the value of each quantity,
if it exists. If it does not exist, explain why.
$\begin{array}{lll}{\text { (a) } \lim _{x \rightarrow 2^{-}} f(x)} & {\text { (b) } \lim _{x \rightarrow 2^{+}} f(x)} & {\text { (c) } \lim _{x \rightarrow 2} f(x)} \\ {\text { (d) } f(2)} & {\text { (e) } \lim _{x \rightarrow 4} f(x)} & {\text { (f) } f(4)}\end{array}$

FUNCTIONS AND LIMITS
The Limit of a Function
Wesley H.
04:03
Essential Calculus Early Transcendentals

$\begin{array}{l}{\text { For the function } f \text { whose graph is given, state the value of }} \\ {\text { each quantity, if it exists. If it does not exist, explain why. }} \\ {\text { (a) } \lim _{x \rightarrow 1} f(x)} & {\text { (b) } \lim _{x \rightarrow 3^{-}} f(x)} \\ {\text { (d) } \lim _{x \rightarrow 3} f(x)} & {\text { (e) } f(3)}\end{array}$

FUNCTIONS AND LIMITS
The Limit of a Function
Wesley H.
07:59
Essential Calculus Early Transcendentals

If a rock is thrown upward on the planet Mars with a velocity of $10 \mathrm{m} / \mathrm{s},$ its height in meters $t$ seconds later is given by $y=10 t-1.86 t^{2}$
$$\text{ (a) Find the average velocity over the given time intervals:}$$
$$
\begin{array}{ll}{\text { (i) }[1,2]} & {\text { (ii) }[1,1.5]} & {\text { (iii) } [1, 1.1]} \\ {\text { (iv) }[1,1.01]} & {\text { (v) }[1,1.001]}\end{array}
$$
$$\text{ (b) Estimate the instantaneous velocity when $t=1$ }$$

FUNCTIONS AND LIMITS
The Limit of a Function
Wesley H.
05:40
Essential Calculus Early Transcendentals

If a ball is thrown into the air with a velocity of 40 $\mathrm{ft} / \mathrm{s}$ , its height in feet $t$ seconds later is given by $y=40 t-16 t^{2}$ .
$$
\begin{array}{l}{\text { (a) Find the average velocity for the time period beginning }} \\ {\text { when } t=2 \text { and lasting }}\end{array}
$$
$$
\begin{array}{ll}{\text { (i) } 0.5 \text { second }} & {\text { (ii) } 0.1 \text { second }} \\ {\text { (iii) } 0.05 \text { second }} & {\text { (iv) } 0.01 \text { second }}\end{array}
$$
$$\text{ (b) Estimate the instantaneous velocity when $t=2$ }$$

FUNCTIONS AND LIMITS
The Limit of a Function
Wesley H.
11:09
Calculus; Graphical, Numerical, Algebraic

Limits and Geometry Let $P \left( a , a ^ { 2 } \right)$ be a point on the parabola $y = x ^ { 2 } , a > 0 .$ Let $O$ be the origin and $( 0 , b )$ the $y$ -intercept of the perpendicular bisector of line segment $O P .$ Find $\lim _ { P \rightarrow O } b$

Limits and Continuity
Rates of Change and Limits
03:41
Calculus; Graphical, Numerical, Algebraic

Controlling Outputs Let $f ( x ) = \sqrt { 3 x - 2 }$
(a) Show that $\lim _ { x \rightarrow 2 } f ( x ) = 2 = f ( 2 )$
(b) Use a graph to estimate values for $a$ and $b$ so that $$1.8 < f ( x ) < 2.2$$ provided $$a < x < b$$
(c) Use a graph to estimate values for $a$ and $b$ so that $$1.99 < f ( x ) < 2.01$$ provided $$a < x < b$$

Limits and Continuity
Rates of Change and Limits
01:30
Calculus; Graphical, Numerical, Algebraic

In Exercises $71 - 74 ,$ complete the following tables and state what you believe $\lim _ { x \rightarrow 0 } f ( x )$ to be.
$$\begin{array} { c | c c c c c } { x } & { - 0.1 } & { - 0.01 } & { - 0.001 } & { - 0.0001 } & { \dots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \end{array}$$
$$\begin{array} { c c c c c } { \text { (b) } } & { 0.1 } & { 0.01 } & { 0.001 } & { 0.0001 } & { \ldots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \\ \hline \end{array}$$
$$f ( x ) = x \sin ( \ln | x | )$$

Limits and Continuity
Rates of Change and Limits
02:26
Calculus; Graphical, Numerical, Algebraic

In Exercises $71 - 74 ,$ complete the following tables and state what you believe $\lim _ { x \rightarrow 0 } f ( x )$ to be.
$$\begin{array} { c | c c c c c } { x } & { - 0.1 } & { - 0.01 } & { - 0.001 } & { - 0.0001 } & { \dots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \end{array}$$
$$\begin{array} { c c c c c } { \text { (b) } } & { 0.1 } & { 0.01 } & { 0.001 } & { 0.0001 } & { \ldots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \\ \hline \end{array}$$
$$f ( x ) = \frac { 10 ^ { x } - 1 } { x }$$

Limits and Continuity
Rates of Change and Limits
Alex J.
04:29
Calculus; Graphical, Numerical, Algebraic

In Exercises $71 - 74 ,$ complete the following tables and state what you believe $\lim _ { x \rightarrow 0 } f ( x )$ to be.
$$\begin{array} { c | c c c c c } { x } & { - 0.1 } & { - 0.01 } & { - 0.001 } & { - 0.0001 } & { \dots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \end{array}$$
$$\begin{array} { c c c c c } { \text { (b) } } & { 0.1 } & { 0.01 } & { 0.001 } & { 0.0001 } & { \ldots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \\ \hline \end{array}$$
$$f ( x ) = \sin \frac { 1 } { x }$$

Limits and Continuity
Rates of Change and Limits
01:19
Calculus; Graphical, Numerical, Algebraic

In Exercises $71 - 74 ,$ complete the following tables and state what you believe $\lim _ { x \rightarrow 0 } f ( x )$ to be.

$$\begin{array} { c | c c c c c } { x } & { - 0.1 } & { - 0.01 } & { - 0.001 } & { - 0.0001 } & { \dots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \end{array}$$

$$\begin{array} { c c c c c } { \text { (b) } } & { 0.1 } & { 0.01 } & { 0.001 } & { 0.0001 } & { \ldots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \\ \hline \end{array}$$

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

Limits and Continuity
Rates of Change and Limits
Alex J.
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