Exam 3: Limits and Continuity

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ occur frequently in calculus. Evaluate this limit for the given value of $x$ and function $f$ . - $( x ) = \frac { x } { 5 } + 7 , x = 10$

Find the limit, if it exists. - $\lim _ { x \rightarrow 1 } \frac { x ^ { 2 } + 8 x - 9 } { x ^ { 2 } - 1 }$

Provide an appropriate response. - $\text { Let } \lim _ { x \rightarrow 10 } f ( x ) = 1024 \text {. Find } \lim _ { x \rightarrow 10 } \sqrt [ 5 ] { f ( x ) }$

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 0 } \frac { x } { \sin 3 x }$

Find the limit. - $\lim _ { x \rightarrow 5 ^ { + } } \frac { 3 } { x ^ { 2 } - 25 }$

Use the graph to evaluate the limit. - $\lim _ { x \rightarrow \theta } f ( x )$

Find the limit, if it exists. - $\lim _ { x \rightarrow - 1 } \frac { 3 x ^ { 2 } + 7 x - 2 } { 3 x ^ { 2 } - 4 x - 2 }$

Find the limit. - $\lim _ { x \rightarrow \infty } \left( \sqrt { 7 x ^ { 2 } + 3 } - \sqrt { 7 x ^ { 2 } - 3 } \right)$

Use the graph to estimate the specified limit. - $\text { Find } \lim _ { x \rightarrow ( - 1 ) ^ { - } } f ( x ) \text { and } \lim _ { x + 1 ) ^ { + } } f ( x )$

Find all points where the function is discontinuous. -

Find the limit. - $\lim _ { x \rightarrow 2 } \left( 3 x ^ { 5 } - 2 x ^ { 4 } - 4 x ^ { 3 } + x ^ { 2 } - 5 \right)$

Use the graph to evaluate the limit. - $\lim _{x \rightarrow \theta} f(x)$

Find the limit if it exists. - $\lim _ { x \rightarrow 8 } \left( 3 x ^ { 2 } - 3 x - 10 \right)$

Use the graph to evaluate the limit. - $\lim _{x \rightarrow 1} f(x)$

$\text { Use the graph to find a } \delta > 0 \text { such that for all } x , 0 < | x - c | < \delta \Rightarrow | f ( x ) - L | < \varepsilon \text {. }$ -

Determine the limit by sketching an appropriate graph. - $\lim _ { x \rightarrow 2 ^ { - } } f ( x ) , \text { where } f ( x ) = \left\{ \begin{array} { l l } - 5 x + 3 & \text { for } x < 2 \\2 x + 4 & \text { for } x \geq 2\end{array} \right.$

Find the limit. - $\lim _ { x \rightarrow } \sqrt { x ^ { 2 } + 2 x } - \sqrt { x ^ { 2 } - 8 x }$

Find all points where the function is discontinuous. -

Find the limit. - $\lim _ { x \rightarrow 1 ^ { + } } \left( \frac { 1 } { x ^ { 4 / 5 } } - \frac { 1 } { ( x - 1 ) ^ { 2 / 5 } } \right)$

Find the average rate of change of the function over the given interval. - $y = \sqrt { 2 x } , [ 2,8 ]$