Question 1

Find the following limit, if it exists.
$$\lim _ { x \rightarrow - 3 } \frac { 9 - 6 x - 3 x ^ { 2 } } { 3 + x }$$

Accepted Answer

A)  $- 6$ 
B)  12 
C)  6 
D)  $- 12$ 
E)  The limit does not exist. 
A)  $- 6$ 
B)  12 
C)  6 
D)  $- 12$ 
E)  The limit does not exist.

Question 2

Use the limit process to find the area of the region between $f ( x ) = x ^ { 2 } + 5$ and the $x$-axis on the interval $[ 0,6 ]$.

Accepted Answer

A)  246 
B)  72 
C)  221 
D)  102 
E)  77 
A)  246 
B)  72 
C)  221 
D)  102 
E)  77

Question 3

Find the following limit, if it exists.
$$\lim _ { x \rightarrow \infty } \frac { 5 } { x ^ { 8 } }$$

Accepted Answer

A)  $\frac { 5 } { 8 }$ 
B)  $- 5$ 
C)  $- 1$ 
D)  0 
E)  The limit does not exist. 
A)  $\frac { 5 } { 8 }$ 
B)  $- 5$ 
C)  $- 1$ 
D)  0 
E)  The limit does not exist.

Question 4

Find $\lim _ { x \rightarrow 13 ^ { - } } \frac { x - 13 } { x ^ { 2 } - 169 }$.

Accepted Answer

A)  $\frac { 1 } { 26 }$ 
B)  $\frac { 1 } { 13 }$ 
C)  $\frac { 13 } { 2 }$ 
D)  $\frac { 2 } { 13 }$ 
E)  limit does not exist 
A)  $\frac { 1 } { 26 }$ 
B)  $\frac { 1 } { 13 }$ 
C)  $\frac { 13 } { 2 }$ 
D)  $\frac { 2 } { 13 }$ 
E)  limit does not exist

Question 5

Use the function below and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal.
$$f ( x ) = 3 x ^ { 4 } - 6 x ^ { 2 } , f ^ { \prime } ( x ) = 12 x ^ { 3 } - 12 x$$

Accepted Answer

A)  $( - 1 , - 3 ) , ( 0,0 ) , ( 1 , - 3 )$ 
B)  $( - 1,3 ) , ( 1,12 )$ 
C)  $( - 1 , - 3 ) , ( 1 , - 3 )$ 
D)  $( 0,0 ) , ( 1,0 )$ 
E)  $( 0,0 ) , ( 1 , - 3 )$ 
A)  $( - 1 , - 3 ) , ( 0,0 ) , ( 1 , - 3 )$ 
B)  $( - 1,3 ) , ( 1,12 )$ 
C)  $( - 1 , - 3 ) , ( 1 , - 3 )$ 
D)  $( 0,0 ) , ( 1,0 )$ 
E)  $( 0,0 ) , ( 1 , - 3 )$

Question 6

Use the limit process to find the slope of the graph of $\sqrt { x + 21 }$ at $( 4,5 )$.

Accepted Answer

A)  $\frac { 1 } { 5 }$ 
B)  5 
C)  $\infty$ 
D)  $\frac { 1 } { 10 }$ 
E)  the slope is undefined at this point 
A)  $\frac { 1 } { 5 }$ 
B)  5 
C)  $\infty$ 
D)  $\frac { 1 } { 10 }$ 
E)  the slope is undefined at this point

Question 7

Use the figure below to approximate the slope of the curve at the point $( x , y )$.

Accepted Answer

A)  1 
B)  $\frac { 1 } { 2 }$ 
C)  $- 1$ 
D)  0 
E)  The slope is undefined. 
A)  1 
B)  $\frac { 1 } { 2 }$ 
C)  $- 1$ 
D)  0 
E)  The slope is undefined.

Question 8

Find $\lim _ { x \rightarrow 15 ^ { - } } \frac { x - 15 } { x ^ { 2 } - 225 }$.

Accepted Answer

A)  $\frac { 1 } { 30 }$ 
B)  $\frac { 1 } { 15 }$ 
C)  $\frac { 15 } { 2 }$ 
D)  $\frac { 2 } { 15 }$ 
E)  limit does not exist 
A)  $\frac { 1 } { 30 }$ 
B)  $\frac { 1 } { 15 }$ 
C)  $\frac { 15 } { 2 }$ 
D)  $\frac { 2 } { 15 }$ 
E)  limit does not exist

Question 9

Complete the table and use the result to estimate
$$\lim _ { x \rightarrow - 2 } \frac { x + 2 } { x ^ { 2 } - x - 6 }$$
numerically.
$\begin{array}{|c|c|c|c|c|c|c|c|}
\hline x & -2.1 & -2.01 & -2.001 & -2 & -1.999 & -1.99 & -1.9 \
\hline f(x) & & & & ? & & & \
\hline
\end{array}$

Accepted Answer

A)  $- \frac { 1 } { 5 }$ 
B)  5 
C)  $\infty$ 
D)  $- 5$ 
E)  limit does not exist 
A)  $- \frac { 1 } { 5 }$ 
B)  5 
C)  $\infty$ 
D)  $- 5$ 
E)  limit does not exist

Question 10

If $f ( x ) = - 2 x ^ { 2 } + x$, find the following limit, if it exists.
$$\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$$

Accepted Answer

A)  $- 4 x + 1$ 
B)  $- 2 x + 1$ 
C)  $2 x + 1$ 
D)  $- 4 x$ 
E)  The limit does not exist. 
A)  $- 4 x + 1$ 
B)  $- 2 x + 1$ 
C)  $2 x + 1$ 
D)  $- 4 x$ 
E)  The limit does not exist.

Question 11

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 8 } \left( x ^ { 2 } + 8 x \right)$ and the $x$-axis on the interval $[ 1,8 ]$.

Accepted Answer

A)  $\frac { 1087 } { 2 }$ 
B)  $\frac { 2023 } { 24 }$ 
C)  $\frac { 191 } { 2 }$ 
D)  574 
E)  $\frac { 1267 } { 24 }$ 
A)  $\frac { 1087 } { 2 }$ 
B)  $\frac { 2023 } { 24 }$ 
C)  $\frac { 191 } { 2 }$ 
D)  574 
E)  $\frac { 1267 } { 24 }$

Question 12

Rewrite $\sum _ { i = 1 } ^ { n } \left[ \frac { 12 } { n } + \left( \frac { 11 i } { n ^ { 2 } } \right) \right] \left( \frac { 8 i } { n } \right)$ as a rational function $S ( n )$ and find $\lim _ { n \rightarrow \infty } S ( n )$.

Accepted Answer

A)  $S ( n ) = \frac { 4 [ 36 n + 11 ( n + 1 ) ] } { 3 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 232 } { 3 }$ 
B)  $S ( n ) = \frac { 4 [ 36 n ( n + 1 ) + 11 ( n + 1 ) ( 2 n + 1 ) ] } { 3 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 232 } { 3 }$ 
C)  $S ( n ) = \frac { 4 [ 36 n ( n + 1 ) + 11 ( n + 1 ) ( 2 n + 1 ) ] } { 3 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 5 } { 6 }$ 
D)  $S ( n ) = \frac { 4 [ 36 + 11 n ] } { 3 n } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 232 } { 3 }$ 
E)  $S ( n ) = \frac { 4 [ 36 n ( n + 1 ) + 11 ( n + 1 ) ( 2 n + 1 ) ] } { 3 n ^ { 2 } }$, limit does not exist 
A)  $S ( n ) = \frac { 4 [ 36 n + 11 ( n + 1 ) ] } { 3 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 232 } { 3 }$ 
B)  $S ( n ) = \frac { 4 [ 36 n ( n + 1 ) + 11 ( n + 1 ) ( 2 n + 1 ) ] } { 3 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 232 } { 3 }$ 
C)  $S ( n ) = \frac { 4 [ 36 n ( n + 1 ) + 11 ( n + 1 ) ( 2 n + 1 ) ] } { 3 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 5 } { 6 }$ 
D)  $S ( n ) = \frac { 4 [ 36 + 11 n ] } { 3 n } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 232 } { 3 }$ 
E)  $S ( n ) = \frac { 4 [ 36 n ( n + 1 ) + 11 ( n + 1 ) ( 2 n + 1 ) ] } { 3 n ^ { 2 } }$, limit does not exist

Question 13

Find the following limit, if it exists.
$$\lim _ { x \rightarrow \infty } \frac { 4 x ^ { 2 } + 3 } { - 2 - x ^ { 2 } }$$

Accepted Answer

A)  $- \frac { 7 } { 3 }$ 
B)  $- 4$ 
C)  4 
D)  7 
E)  The limit does not exist. 
A)  $- \frac { 7 } { 3 }$ 
B)  $- 4$ 
C)  4 
D)  7 
E)  The limit does not exist.

Question 14

Use the graph to determine $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 3 x } { x }$ (if it exists).

Accepted Answer

A)  0 
B)  3 
C)  $- 3$ 
D)  6 
E)  limit does not exist 
A)  0 
B)  3 
C)  $- 3$ 
D)  6 
E)  limit does not exist

Question 15

Use the limit process to find the area of the region between $f ( x ) = 6 x + 7$ and the $x$-axis on the interval $[ 0,5 ]$.

Accepted Answer

A)  185 
B)  110 
C)  65 
D)  75 
E)  82 
A)  185 
B)  110 
C)  65 
D)  75 
E)  82

Question 16

Find $\lim _ { x \rightarrow \infty } \frac { 1 - 3 x } { 1 + 7 x }$ (if it exists).

Accepted Answer

A)  0 
B)  $- \frac { 3 } { 7 }$ 
C)  $\frac { 3 } { 7 }$ 
D)  $- \infty$ 
E)  limit does not exist 
A)  0 
B)  $- \frac { 3 } { 7 }$ 
C)  $\frac { 3 } { 7 }$ 
D)  $- \infty$ 
E)  limit does not exist

Question 17

Estimate the following limit numerically, if it exists.
$$\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 5 x - 6 }$$

Accepted Answer

A)  $\frac { 1 } { 7 }$ 
B)  0 
C)  $- \frac { 1 } { 5 }$ 
D)  $\frac { 1 } { 5 }$ 
E)  The limit does not exist. 
A)  $\frac { 1 } { 7 }$ 
B)  0 
C)  $- \frac { 1 } { 5 }$ 
D)  $\frac { 1 } { 5 }$ 
E)  The limit does not exist.

Question 18

Complete the table and use the result to estimate
$$\lim _ { x \rightarrow - 8 } \frac { x + 8 } { x ^ { 2 } + 4 x - 32 }$$
numerically.
$\begin{array}{|c|c|c|c|c|c|c|c|}
\hline x & -8.1 & -8.01 & -8.001 & -8 & -7.999 & -7.99 & -7.9 \
\hline f(x) & & & & ? & & & \
\hline
\end{array}$

Accepted Answer

A)  $- \frac { 1 } { 12 }$ 
B)  12 
C)  $\infty$ 
D)  $- 12$ 
E)  limit does not exist 
A)  $- \frac { 1 } { 12 }$ 
B)  12 
C)  $\infty$ 
D)  $- 12$ 
E)  limit does not exist

Question 19

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 6 }$.

Accepted Answer

A)  $\frac { 1 } { 6 \sqrt { x - 6 } }$ 
B)  $\frac { 1 } { 2 \sqrt { x - 6 } }$ 
C)  $\frac { \sqrt { x - 6 } } { 2 }$ 
D)  $\sqrt { x - 6 }$ 
E)  limit does not exist 
A)  $\frac { 1 } { 6 \sqrt { x - 6 } }$ 
B)  $\frac { 1 } { 2 \sqrt { x - 6 } }$ 
C)  $\frac { \sqrt { x - 6 } } { 2 }$ 
D)  $\sqrt { x - 6 }$ 
E)  limit does not exist

Question 20

Find $\lim _ { x \rightarrow 9 ^ { - } } \frac { x - 9 } { x ^ { 2 } - 81 }$.

Accepted Answer

A)  $\frac { 1 } { 18 }$ 
B)  $\frac { 1 } { 9 }$ 
C)  $\frac { 9 } { 2 }$ 
D)  $\frac { 2 } { 9 }$ 
E)  limit does not exist 
A)  $\frac { 1 } { 18 }$ 
B)  $\frac { 1 } { 9 }$ 
C)  $\frac { 9 } { 2 }$ 
D)  $\frac { 2 } { 9 }$ 
E)  limit does not exist

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } \frac { 9 - 6 x - 3 x ^ { 2 } } { 3 + x }$

Use the limit process to find the area of the region between $f ( x ) = x ^ { 2 } + 5$ and the $x$ -axis on the interval $[ 0,6 ]$ .

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 5 } { x ^ { 8 } }$

Find $\lim _ { x \rightarrow 13 ^ { - } } \frac { x - 13 } { x ^ { 2 } - 169 }$ .

Use the function below and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal. $f ( x ) = 3 x ^ { 4 } - 6 x ^ { 2 } , f ^ { \prime } ( x ) = 12 x ^ { 3 } - 12 x$

Use the limit process to find the slope of the graph of $\sqrt { x + 21 }$ at $( 4,5 )$ .

Use the figure below to approximate the slope of the curve at the point $( x , y )$ .

Find $\lim _ { x \rightarrow 15 ^ { - } } \frac { x - 15 } { x ^ { 2 } - 225 }$ .

Complete the table and use the result to estimate $\lim _ { x \rightarrow - 2 } \frac { x + 2 } { x ^ { 2 } - x - 6 }$ numerically. x -2.1 -2.01 -2.001 -2 -1.999 -1.99 -1.9 f(x) ?

If $f ( x ) = - 2 x ^ { 2 } + x$ , find the following limit, if it exists. $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 8 } \left( x ^ { 2 } + 8 x \right)$ and the $x$ -axis on the interval $[ 1,8 ]$ .

Rewrite $\sum _ { i = 1 } ^ { n } \left[ \frac { 12 } { n } + \left( \frac { 11 i } { n ^ { 2 } } \right) \right] \left( \frac { 8 i } { n } \right)$ as a rational function $S ( n )$ and find $\lim _ { n \rightarrow \infty } S ( n )$ .

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 4 x ^ { 2 } + 3 } { - 2 - x ^ { 2 } }$

Use the graph to determine $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 3 x } { x }$ (if it exists). $Use the graph to determine \lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 3 x } { x } (if it exists).$

Use the limit process to find the area of the region between $f ( x ) = 6 x + 7$ and the $x$ -axis on the interval $[ 0,5 ]$ .

Find $\lim _ { x \rightarrow \infty } \frac { 1 - 3 x } { 1 + 7 x }$ (if it exists).

Estimate the following limit numerically, if it exists. $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 5 x - 6 }$

Complete the table and use the result to estimate $\lim _ { x \rightarrow - 8 } \frac { x + 8 } { x ^ { 2 } + 4 x - 32 }$ numerically. x -8.1 -8.01 -8.001 -8 -7.999 -7.99 -7.9 f(x) ?

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 6 }$ .

Find $\lim _ { x \rightarrow 9 ^ { - } } \frac { x - 9 } { x ^ { 2 } - 81 }$ .

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Filters

Exam 11: Limits and an Introduction to Calculus

Find the following limit, if it exists. lim⁡x→−39−6x−3x23+x\lim _ { x \rightarrow - 3 } \frac { 9 - 6 x - 3 x ^ { 2 } } { 3 + x }limx→−3​3+x9−6x−3x2​

Use the limit process to find the area of the region between f(x)=x2+5f ( x ) = x ^ { 2 } + 5f(x)=x2+5 and the xxx -axis on the interval [0,6][ 0,6 ][0,6] .

Find the following limit, if it exists. lim⁡x→∞5x8\lim _ { x \rightarrow \infty } \frac { 5 } { x ^ { 8 } }limx→∞​x85​

Find lim⁡x→13−x−13x2−169\lim _ { x \rightarrow 13 ^ { - } } \frac { x - 13 } { x ^ { 2 } - 169 }limx→13−​x2−169x−13​ .

Use the function below and its derivative to determine any points on the graph of fff at which the tangent line is horizontal. f(x)=3x4−6x2,f′(x)=12x3−12xf ( x ) = 3 x ^ { 4 } - 6 x ^ { 2 } , f ^ { \prime } ( x ) = 12 x ^ { 3 } - 12 xf(x)=3x4−6x2,f′(x)=12x3−12x

Use the limit process to find the slope of the graph of x+21\sqrt { x + 21 }x+21​ at (4,5)( 4,5 )(4,5) .

Use the figure below to approximate the slope of the curve at the point (x,y)( x , y )(x,y) .

Find lim⁡x→15−x−15x2−225\lim _ { x \rightarrow 15 ^ { - } } \frac { x - 15 } { x ^ { 2 } - 225 }limx→15−​x2−225x−15​ .

Complete the table and use the result to estimate lim⁡x→−2x+2x2−x−6\lim _ { x \rightarrow - 2 } \frac { x + 2 } { x ^ { 2 } - x - 6 }limx→−2​x2−x−6x+2​ numerically. x -2.1 -2.01 -2.001 -2 -1.999 -1.99 -1.9 f(x) ?

If f(x)=−2x2+xf ( x ) = - 2 x ^ { 2 } + xf(x)=−2x2+x , find the following limit, if it exists. lim⁡h→0f(x+h)−f(x)h\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }limh→0​hf(x+h)−f(x)​

Use the limit process to find the area of the region between f(x)=18(x2+8x)f ( x ) = \frac { 1 } { 8 } \left( x ^ { 2 } + 8 x \right)f(x)=81​(x2+8x) and the xxx -axis on the interval [1,8][ 1,8 ][1,8] .

Find the following limit, if it exists. lim⁡x→∞4x2+3−2−x2\lim _ { x \rightarrow \infty } \frac { 4 x ^ { 2 } + 3 } { - 2 - x ^ { 2 } }limx→∞​−2−x24x2+3​

Use the graph to determine lim⁡x→0x2−3xx\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 3 x } { x }limx→0​xx2−3x​ (if it exists).

Use the limit process to find the area of the region between f(x)=6x+7f ( x ) = 6 x + 7f(x)=6x+7 and the xxx -axis on the interval [0,5][ 0,5 ][0,5] .

Find lim⁡x→∞1−3x1+7x\lim _ { x \rightarrow \infty } \frac { 1 - 3 x } { 1 + 7 x }limx→∞​1+7x1−3x​ (if it exists).

Estimate the following limit numerically, if it exists. lim⁡x→1x−1x2+5x−6\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 5 x - 6 }limx→1​x2+5x−6x−1​

Complete the table and use the result to estimate lim⁡x→−8x+8x2+4x−32\lim _ { x \rightarrow - 8 } \frac { x + 8 } { x ^ { 2 } + 4 x - 32 }limx→−8​x2+4x−32x+8​ numerically. x -8.1 -8.01 -8.001 -8 -7.999 -7.99 -7.9 f(x) ?

Find lim⁡h→0f(x+h)−f(x)h\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }limh→0​hf(x+h)−f(x)​ for f(x)=x−6f ( x ) = \sqrt { x - 6 }f(x)=x−6​ .

Find lim⁡x→9−x−9x2−81\lim _ { x \rightarrow 9 ^ { - } } \frac { x - 9 } { x ^ { 2 } - 81 }limx→9−​x2−81x−9​ .

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Filters

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } \frac { 9 - 6 x - 3 x ^ { 2 } } { 3 + x }$

Use the limit process to find the area of the region between $f ( x ) = x ^ { 2 } + 5$ and the $x$ -axis on the interval $[ 0,6 ]$ .

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 5 } { x ^ { 8 } }$

Find $\lim _ { x \rightarrow 13 ^ { - } } \frac { x - 13 } { x ^ { 2 } - 169 }$ .

Use the function below and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal. $f ( x ) = 3 x ^ { 4 } - 6 x ^ { 2 } , f ^ { \prime } ( x ) = 12 x ^ { 3 } - 12 x$

Use the limit process to find the slope of the graph of $\sqrt { x + 21 }$ at $( 4,5 )$ .

Use the figure below to approximate the slope of the curve at the point $( x , y )$ .

Find $\lim _ { x \rightarrow 15 ^ { - } } \frac { x - 15 } { x ^ { 2 } - 225 }$ .

Complete the table and use the result to estimate $\lim _ { x \rightarrow - 2 } \frac { x + 2 } { x ^ { 2 } - x - 6 }$ numerically. x -2.1 -2.01 -2.001 -2 -1.999 -1.99 -1.9 f(x) ?

If $f ( x ) = - 2 x ^ { 2 } + x$ , find the following limit, if it exists. $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 8 } \left( x ^ { 2 } + 8 x \right)$ and the $x$ -axis on the interval $[ 1,8 ]$ .

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 4 x ^ { 2 } + 3 } { - 2 - x ^ { 2 } }$

Use the graph to determine $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 3 x } { x }$ (if it exists). $Use the graph to determine \lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 3 x } { x } (if it exists).$

Use the limit process to find the area of the region between $f ( x ) = 6 x + 7$ and the $x$ -axis on the interval $[ 0,5 ]$ .

Find $\lim _ { x \rightarrow \infty } \frac { 1 - 3 x } { 1 + 7 x }$ (if it exists).

Estimate the following limit numerically, if it exists. $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 5 x - 6 }$

Complete the table and use the result to estimate $\lim _ { x \rightarrow - 8 } \frac { x + 8 } { x ^ { 2 } + 4 x - 32 }$ numerically. x -8.1 -8.01 -8.001 -8 -7.999 -7.99 -7.9 f(x) ?

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 6 }$ .

Find $\lim _ { x \rightarrow 9 ^ { - } } \frac { x - 9 } { x ^ { 2 } - 81 }$ .