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Limits of a Function

Calculus 1 / Ab

610 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.
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
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
01:00
Calculus; Graphical, Numerical, Algebraic

In Exercises $67 - 70$ , use the following function.

$f ( x ) = \left\{ \begin{array} { l l } { 2 - x , } & { x \leq 1 } \\ { \frac { x } { 2 } + 1 , } & { x > 1 } \end{array} \right.$

Multiple Choice What is the value of $f ( 1 ) ?$

(A) 5$/ 2 \quad$ (B) 3$/ 2 \quad$ (C) (D) 0 1(E) does not exist

Limits and Continuity
Rates of Change and Limits
01:26
Calculus; Graphical, Numerical, Algebraic

In Exercises $67 - 70$ , use the following function.

$f ( x ) = \left\{ \begin{array} { l l } { 2 - x , } & { x \leq 1 } \\ { \frac { x } { 2 } + 1 , } & { x > 1 } \end{array} \right.$

Multiple Choice What is the value of $\lim _ { x \rightarrow 1 } + f ( x ) ?$

(A) 5$/ 2$ $( \mathrm { B } ) 3 / 2$ $( \mathbf { C } ) 1$ $( \mathbf { D } ) 0$ (E) does not exist

Limits and Continuity
Rates of Change and Limits
01:54
Calculus; Graphical, Numerical, Algebraic

True or False $\lim _ { x \rightarrow 0 } \frac { x + \sin x } { x } = 2 .$ Justify your answer.

Limits and Continuity
Rates of Change and Limits
05:24
Calculus; Graphical, Numerical, Algebraic

Free Fall on a Small Airless Planet A rock released from rest to fall on a small airless planet falls $y = g t ^ { 2 } \mathrm { m }$ in $t \mathrm { sec } , g$ a constant. Suppose that the rock falls to the bottom of a crevasse 20$\mathrm { m }$ below and reaches the bottom in 4$\mathrm { sec. }$

(a) Find the value of $g .$
(b) Find the average speed for the fall.
(c) With what speed did the rock hit the bottom?

Limits and Continuity
Rates of Change and Limits
03:18
Calculus; Graphical, Numerical, Algebraic

In Exercises $59 - 62 ,$ find the limit graphically. Use the Sandwich Theorem to confirm your answer.
$$\lim _ { x \rightarrow 0 } x ^ { 2 } \cos \frac { 1 } { x ^ { 2 } }$$

Limits and Continuity
Rates of Change and Limits
02:28
Calculus; Graphical, Numerical, Algebraic

In Exercises $59 - 62 ,$ find the limit graphically. Use the Sandwich Theorem to confirm your answer.
$$\lim _ { x \rightarrow 0 } x ^ { 2 } \sin x$$

Limits and Continuity
Rates of Change and Limits
05:10
Calculus; Graphical, Numerical, Algebraic

In Exercises $55 - 58 ,$ complete parts $( a ) - ( d )$ for the piecewise-definedfunction. $\quad ($ a) Draw the graph of $f$ .
(b) At what points $c$ in the domain of $f$ does $\lim _ { x \rightarrow c } f ( x )$ exist?
(c) At what points $c$ does only the left-hand limit exist?
(d) At what points $c$ does only the right-hand limit exist?
$$f ( x ) = \left\{ \begin{array} { l l } { x , } & { - 1 \leq x < 0 , \text { or } 0 < x \leq 1 } \\ { 1 , } & { x = 0 } \\ { 0 , } & { x < - 1 , \text { or } x > 1 } \end{array} \right.$$

Limits and Continuity
Rates of Change and Limits
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