Exam 3: Limits and Continuity

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit. - $\lim _ { x \rightarrow 0 } \frac { \sqrt { 10 + 10 x } - \sqrt { 10 } } { x }$

Use the graph to evaluate the limit. - $\lim _{x \rightarrow 0} f(x)$

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

Find the limit. - $\lim _ { x \rightarrow 1 ^ { - } } \left( \frac { 1 } { x ^ { 4 / 5 } } - \frac { 1 } { ( x - 1 ) ^ { 3 / 5 } } \right)$

For the function f whose graph is given, determine the limit. -Find $\lim _ { x - 4 ^ { + } } f ( x )$ .

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 0 } \frac { \sin 5 x } { x }$

Find the average rate of change of the function over the given interval. - $y = 5 x ^ { 3 } - 8 x ^ { 2 } - 3 , [ - 9,5 ]$

Provide an appropriate response. -Let $\lim _ { x \rightarrow 6 } f ( x ) = 9$ and $\lim _ { x \rightarrow 6 } g ( x ) = - 7$ . Find $\lim _ { x \rightarrow 6 } \frac { f ( x ) } { g ( x ) }$ .

Find the limit, if it exists. - $\lim _ { x \rightarrow } \frac { x ^ { 2 } - 2 x - 15 } { x + 3 }$

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

Find the limit. - $\lim _ { x \rightarrow 2 } \frac { 1 } { x + 2 }$

Find the intervals on which the function is continuous. - $y = \frac { 3 } { x + 7 } - 3 x$

Use the table to estimate the rate of change of y at the specified value of x. - $x= 1 .$ x 0 0 .2 0.02 .4 0.08 .6 0.18 .8 0.32 0 0.5 .2 0.72 .4 0.98

Find the limit. - $\lim _ { h \rightarrow 0 ^ { - } } \frac { \sqrt { 13 } - \sqrt { 8 h ^ { 2 } + 9 h + 13 } } { h }$

Determine if the given function can be extended to a continuous function at x = 0. If so, approximate the extended function's value at x = 0 (rounded to four decimal places if necessary). If not, determine whether the function can be continuously extended from the left or from the right and provide the values of the extended functions at x = 0. Otherwise write "no continuous extension." - $f ( x ) = \frac { \cos 2 x } { | 2 x | }$

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 2 x + \sin x } { x }$

Find the limit. - $\lim _ { x \rightarrow \infty } \frac { 8 x ^ { 3 } - 2 x ^ { 2 } + 3 x } { - x ^ { 3 } - 2 x + 6 }$

Determine the limit by sketching an appropriate graph. - $\lim _ { x \rightarrow 4 ^ { + } } f ( x ) , \text { where } f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } + 6 & \text { for } x \neq - 4 \\0 & \text { for } x = - 4\end{array} \right.$

Find the limit. - $\lim _ { x \rightarrow 2 } \frac { f ( x ) } { x ^ { 2 } } = 4$ , find $\lim _ { x \rightarrow 2 } \frac { f ( x ) } { x }$ .

Find all points where the function is discontinuous. -