Exam 3: Limits and Continuity

Sketch the graph of a function y = f(x) that satisfies the given conditions. - $\lim _ { x \rightarrow } f ( x ) = 0 , \quad \lim _ { x - \theta ^ { + } } f ( x ) = \infty$

Find the intervals on which the function is continuous. - $y = \frac { 5 } { | x | + 2 } - \frac { x ^ { 2 } } { 7 }$

Use the table to estimate the rate of change of y at the specified value of x. -

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit. -Write the formal notation for the principle "the limit of a quotient is the quotient of the limits" and include a statement of any restrictions on the principle.

Find the intervals on which the function is continuous. - $y = \frac { 4 \cos \theta } { \theta + 7 }$

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 0 } \frac { \sin 3 x \cot 4 x } { \cot 5 x }$

Determine the limit by sketching an appropriate graph. - $\lim _ { x \rightarrow 3 ^ { + } } f ( x ) \text {, where } f ( x ) = \left\{ \begin{array} { l l } 3 x & - 3 \leq x < 0 , \text { or } 0 < x \leq 2 \\3 & x = 0 \\0 & x < - 3 \text { or } x > 2\end{array} \right.$

Graph the rational function. Include the graphs and equations of the asymptotes. - $y = \frac { 2 x ^ { 2 } } { 4 - x ^ { 2 } }$

Find the slope of the curve at the given point P and an equation of the tangent line at P. - $y = x ^ { 3 } - 8 x , P ( 1 , - 7 )$

Use the slopes of UQ, UR, US, and UT to estimate the rate of change of y at the specified value of x. - $x = 2.5$

Divide numerator and denominator by the highest power of x in the denominator to find the limit. - $\lim _ { x \rightarrow \infty } \sqrt { \frac { 16 x ^ { 2 } } { 3 + 9 x ^ { 2 } } }$

Graph the rational function. Include the graphs and equations of the asymptotes. - $y = \frac { x - 1 } { x + 1 }$

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

Provide an appropriate response. - $\text { Use the Intermediate Value Theorem to prove that } x ( x - 2 ) ^ { 2 } = 2 \text { has a solution between } 1 \text { and } 3 \text {. }$

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

Find the average rate of change of the function over the given interval. - $y = x ^ { 2 } + 9 x , [ 1,8 ]$

Provide an appropriate response. -Let $\lim _ { x \rightarrow 1 } f ( x ) = - 2$ and $\lim _ { x \rightarrow 1 } g ( x ) = - 9$ . Find $\lim _ { x \rightarrow 1 } [ f ( x ) - g ( x ) ]$ .

Use the slopes of UQ, UR, US, and UT to estimate the rate of change of y at the specified value of x. - $x= 5$

Find the limit if it exists. - $\lim _ { x \rightarrow 5 } ( x + 248 ) ^ { 3 / 5 }$

Find all points where the function is discontinuous. -