Deck 11: Limits of Functions

Question 1

Find $\lim _ { x \rightarrow 8 } \frac { 3 x } { \sqrt { x + 5 } }$ by direct substitution.

Answer 1

$\frac { 24 } { \sqrt { 13 } }$

Question 2

Find the limit by direct substitution.
​ $\lim _ { x \rightarrow - 3 } \left( \frac { 12 x - 10 } { 3 x + 14 } \right)$ ​

Answer 2

$\lim _ { x \rightarrow - 3 } \left( \frac { 12 x - 10 } { 3 x + 14 } \right) = - \frac { 46 } { 5 }$

Question 3

Find the limit (if it exists). Use a graphing utility to verify your result graphically.
​ $\lim _ { x \rightarrow - 4 } \frac { x ^ { 2 } + 10 x + 24 } { x + 4 }$ ​

Answer 3

$$\lim _ { x \rightarrow - 4 } \frac { x ^ { 2 } + 10 x + 24 } { x + 4 } = 2$$ ​
​ ​

Question 4

Find the limit (if it exists).
​ $\lim _ { y \rightarrow \infty } \frac { 6 y ^ { 4 } } { y ^ { 2 } + 4 }$ ​

Question 5

Find the limit.
​ $\lim _ { x \rightarrow 0 } \frac { 1 - e ^ { - 7 x } } { x }$ ​

Question 6

Select the correct function for the graph using oblique asymptotes as aids. ​

Question 7

Find the limit by direct substitution.
​ $\lim _ { x \rightarrow 5 } \frac { 15 x } { \sqrt { x + 4 } }$ ​

Question 8

Use the graph to find $\lim _ { x \rightarrow 0 } 2 \sin \left( \frac { \pi } { 10 x } \right)$ . ​

Question 9

Select the correct graph for the following function using a graphing utility. Determine whether the limit exists or not. $f ( x ) = \sin 5 \pi x$ , $\lim _ { x \rightarrow 2 } f ( x )$

Question 10

Find the limit (if it exists).
​ $\lim _ { x \rightarrow - \infty } \frac { 4 x ^ { 2 } - 4 x - 12 } { 4 - 9 x - 7 x ^ { 2 } }$ ​

Question 11

Complete the table and numerically estimate the limit as x approaches infinity for $f ( x ) = x - \sqrt { x ^ { 2 } + 3 }$ .
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$$\begin{array} { | c | c | c | c | c | c | c | c | } \hline x & 10 ^ { 0 } & 10 ^ { 1 } & 10 ^ { 2 } & 10 ^ { 3 } & 10 ^ { 4 } & 10 ^ { 5 } & 10 ^ { 6 } \\\hline f ( x ) & & & & & & & \\\hline\end{array}$$

Question 12

Use asymptotes to match $f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 2 } + 1 }$ with its graph.

Question 13

Graph the function. Determine the limit (if it exists) by evaluating the corresponding one-sided limits.
​ $$\lim _ { x \rightarrow 5 } \frac { 1 } { x ^ { 2 } + 5 }$$ ​

Question 14

Find $\lim _ { x \rightarrow 5 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 5 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 9 } } { 3 x ^ { 2 } }$ .

Question 15

Find the limit (if it exists). Use a graphing utility to verify your result graphically.
​ $$\lim _ { x \rightarrow 0 } \frac { \frac { 4 } { x + 4 } - 1 } { x }$$