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 $f$ . - $( x ) = - 4 x + 7 , x = 6$

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 $f$ . - $f ( x ) = \sqrt { x } , x = 13$

Divide numerator and denominator by the highest power of x in the denominator to find the limit. - $\lim _ { x \rightarrow \infty } \sqrt { \frac { 25 x ^ { 2 } + x - 3 } { ( x - 11 ) ( x + 1 ) } }$

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

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

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 $x , 0 < | \mathrm { x } - \mathrm { c } | < \delta \Rightarrow | \mathrm { f } ( \mathrm { x } ) - \mathrm { L } | < \varepsilon$ . - $f ( x ) = 4 x + 1 , L = 13 , c = 3$ , and $\varepsilon = 0.01$

Find the limit if it exists. - $\mathrm { m } _ { 27 } \mathrm { x } ^ { 2 / 3 }$

Answer the question. -Is $f$ continuous at $f ( 1 )$ ? $f ( x ) = \left\{ \begin{array} { l l } - x ^ { 2 } + 1 , & - 1 \leq x < 0 \\3 x , & 0 < x < 1 \\- 4 , & x = 1 \\- 3 x + 6 & 1 < x < 3 \\5 , & 3 < x < 5\end{array} \right.$

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

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

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit. - $\lim _ { x \rightarrow 8 } \frac { x ^ { 2 } - 9 } { \sqrt { x ^ { 2 } + 7 } - 4 }$

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

Find numbers a and b, or k, so that f is continuous at every point. - $f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } , & x < - 4 \\a x + b , - 4 \leq x \leq 2 \\x + 2 , & x > 2\end{array} \right.$

Find the limit. - $\lim _ { x \rightarrow 6 } ( 9 x + 5 )$

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

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

Find the limit and determine if the function is continuous at the point being approached. - $\lim _ { x \rightarrow 0 } \sec \left( x \sec ^ { 2 } x - x \tan ^ { 2 } x - 1 \right)$

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

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 timłRipening Time (minutes) (days) 10 4.3 15 3.2 20 2.7 25 2.1 30 1.3 Plot the data and then find a line approximating the data. With the aid of this line, determine the rate of change of Ripening time with respect to exposure time. Round your answer to two significant digits.

Find the intervals on which the function is continuous. - $y = \frac { 2 } { x ^ { 2 } - 25 }$