Exam 12: Limits and An Introduction To Calculus

Use the graph to determine $\lim _ { x \rightarrow 0 } \frac { 8 x ^ { 2 } - 8 x } { x }$ (if it exists).

Select the correct graph of the following function. $f ( x ) = 8 \left( 5 x - \sqrt { 25 x ^ { 2 } + x } \right)$

Use the graph to determine the limit visually (if it exists).Then identify another function g₂(x)that agrees with the given function at all but one point. $g ( x ) = \frac { - 3 x ^ { 2 } + x } { x }$ $\lim _ { x \rightarrow - 2 } g ( x ) =$

Find the derivative of the function. $f ( x ) = \sqrt { x + 3 }$

Find the limit (if it exists).Use a graphing utility to verify your result graphically. $\lim _ { a \rightarrow - 5 } \frac { a ^ { 3 } + 125 } { a + 5 }$

Find $\lim _ { x \rightarrow - 2 } \left( \frac { 1 } { 2 } x ^ { 5 } - 5 x \right)$ by direct substitution.

Find the limit $\lim _ { x \rightarrow 0 } \frac { \tan 5 x } { x }$

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

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point. $f ( x ) = \frac { 1 } { x + 6 } , \quad \left( - 2 , \frac { 1 } { 4 } \right)$

Find $\lim _ { x \rightarrow 6 } \frac { x - 2 } { x ^ { 2 } + 15 x + 54 }$ by direct substitution.

Use the first six terms to predict the limit of the sequence $\alpha _ { n } = \frac { 6 n ^ { 3 } + 5 } { n + 3 }$ (assume n begins with 1).

Find $\lim _ { x \rightarrow 4 } [ f ( x ) g ( x ) ]$ . $f ( x ) = \frac { x } { 8 - x } , g ( x ) = \sin \pi x$

Evaluate the sum using the summation formula and property. $\sum _ { j = 1 } ^ { 20 } \left( j ^ { 2 } + j \right)$

Evaluate the sum using the summation formula and property. $\sum _ { i = 1 } ^ { 60 } 7$

Use the limit process to find the slope of the graph of the function at the specified point.Use a graphing utility to confirm your result. $g ( x ) = \frac { 25 } { x }$

Select the correct graph for the following function and find the limit (if it exists)as x approaches 2. $f ( x ) = \left\{ \begin{array} { l } - 4 x , x \leq 2 \\x ^ { 2 } - 8 x + 1 , x > 2\end{array} \right.$

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not. $f ( x ) = \frac { 10 } { 2 + e ^ { 1 / x } } , \lim _ { x \rightarrow 0 } f ( x )$

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point. $f ( x ) = 6 - x ^ { 2 } , \quad ( 0,6 )$

Use the given information to evaluate the limit. $\lim _ { x \rightarrow c } g ( x ) = 6$ $\lim _ { x \rightarrow c } [ - 6 g ( x ) ]$

Use the position function $s ( t ) = 16 t ^ { 2 } + 109$ to find the velocity in feet/second at time $t = 2$ seconds.The velocity at time $t = c$ seconds is given by $\lim _ { t \rightarrow c } \frac { [ s ( c ) - s ( t ) ] } { ( c - t ) }$ .