Exam 11: Limits of Functions

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

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

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

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

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

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

Complete the table and numerically estimate the limit as x approaches infinity for $f ( x ) = x - \sqrt { x ^ { 2 } + 3 }$ . ​ ​ x 1 1 1 1 1 1 1 f(x)

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

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

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

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

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

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

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

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