Exam 3: Limits and Continuity

Provide an appropriate response. -Use a calculator to graph the function $\mathrm { f }$ to see whether it appears to have a continuous extension to the origin. I it does, use Trace and Zoom to find a good candidate for the extended function's value at $x = 0$ . If the function does not appear to have a continuous extension, can it be extended to be continuous at the origin from the right the left? If so, what do you think the extended function's value(s) should be? $f ( x ) = \frac { 4 \sin x } { | x | }$

A function $\mathrm { f } ( \mathrm { x } )$ , a point $\mathrm { c }$ , the limit of $\mathrm { f } ( \mathrm { x } )$ as $x$ approaches $\mathrm { c }$ , and a positive number $\varepsilon$ is given. Find a number $\delta > 0$ such that for all $x , 0 < | \mathrm { x } - \mathrm { c } | < \delta \Rightarrow | \mathrm { f } ( \mathrm { x } ) - \mathrm { L } | < \varepsilon$ . - $f ( x ) = - 4 x + 9 , L = - 3 , c = 3$ , and $\varepsilon = 0.01$

Find the limit. - $\lim _ { x \rightarrow \theta ^ { + } } ( 1 + \csc x )$

Sketch the graph of a function y = f(x) that satisfies the given conditions. - $f ( 0 ) = 0 , f ( 1 ) = 5 , f ( - 1 ) = - 5 , \lim _ { x \rightarrow \infty } f ( x ) = - 4 , \lim _ { x \rightarrow \infty } f ( x ) = 4 \text {. }$

Find the limit. - $\lim _ { x \rightarrow 0 } ( \sqrt { x } - 2 )$

Solve the problem. -When exposed to ethylene gas, green bananas will ripen at an accelerated rate. The number of days for ripening becomes shorter for longer exposure times. Assume that the table below gives average ripening times of Bananas for several different ethylene exposure times: Exposure timdRipening Time (minutes) (days) 10 4.2 15 3.5 20 2.6 25 2.1 30 1.1 Plot the data and then find a line approximating the data. With the aid of this line, find the limit of the average ripening Time as the exposure time to ethylene approaches 0. Round your answer to the nearest tenth.

For the function f whose graph is given, determine the limit. - $\text { Find } \lim _{x \rightarrow 0} f(x) \text {. }$

Find the slope of the curve at the given point P and an equation of the tangent line at P. - $y = x ^ { 3 } - 2 x ^ { 2 } + 4 , P ( 1,3 )$

Find the limit. - $\lim _ { x \rightarrow \infty } \frac { 6 } { 5 - \left( 6 / x ^ { 2 } \right) }$

Find the limit if it exists. - $\lim _ { x \rightarrow 8 } \sqrt { 2 x + 21 }$

Find the slope of the curve at the given point P and an equation of the tangent line at P. - $y = x ^ { 2 } + 5 x , P ( 4,36 )$

For the function f whose graph is given, determine the limit. -Find $\lim _ { x \rightarrow 1 } f ( x )$ .

Divide numerator and denominator by the highest power of x in the denominator to find the limit. - $\lim _ { t\rightarrow \infty } \frac { \sqrt { 4 t ^ { 2 } - 8 } } { t - 2 }$

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

Find the intervals on which the function is continuous. - $y = \frac { 2 } { ( x + 3 ) ^ { 2 } + 6 }$

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

Provide an appropriate response. -Given $\lim _ { x \rightarrow \theta ^ { - } } f ( x ) = L , \lim _ { x - \theta ^ { + } } f ( x ) = L _ { r }$ , and $L _ { 1 } = L _ { r }$ , which of the following statements is false? I. $\lim _ { x \rightarrow 9 } f ( x ) = L _ { l }$ II. $\lim _ { x \rightarrow \theta } f ( x ) = L _ { r }$ III. $\lim _ { x \rightarrow \theta } f ( x )$ does not exist.

Provide an appropriate response. - $\text { Use the Intermediate Value Theorem to prove that } 5 x ^ { 4 } - 5 x ^ { 3 } + 3 x - 10 = 0 \text { has a solution between } - 2 \text { and } - 1 \text {. }$

A function $\mathrm { f } ( \mathrm { x } )$ , a point $\mathrm { c }$ , the limit of $\mathrm { f } ( \mathrm { x } )$ as $x$ approaches $\mathrm { c }$ , and a positive number $\varepsilon$ is given. Find a number $\delta > 0$ such that for all $x , 0 < | \mathrm { x } - \mathrm { c } | < \delta \Rightarrow | \mathrm { f } ( \mathrm { x } ) - \mathrm { L } | < \varepsilon$ . - $f ( x ) = \frac { x ^ { 2 } + - 16 x + 60 } { x + - 6 } , c = 6 , \varepsilon = 0.02$

Answer the question. -Does $\lim _ { x \rightarrow \mathbf { - } } f ( x )$ exist? $f ( x ) = \left\{ \begin{array} { l l } - x ^ { 2 } + 1 , & - 1 \leq x < 0 \\2 x , & 0 < x < 1 \\- 3 , & x = 1 \\- 2 x + 4 & 1 < x < 3 \\4 , & 3 < x < 5\end{array} \right.$