Exam 3: Limits and Continuity

Provide an appropriate response. -Give an example of a function f(x) that is continuous for all values of x except x = 10, where it has a nonremovable discontinuity. Explain how you know that f is discontinuous at x = 10 and why the discontinuity is nonremovable.

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

Find the limit. - $\lim _ { x \rightarrow 0 ^ { + } } \left( \frac { 1 } { x ^ { 1 / 5 } } + 8 \right)$

Sketch the graph of a function y = f(x) that satisfies the given conditions. - $f ( 0 ) = 0 , \lim _ { x \rightarrow \infty } f ( x ) = 0 , \lim _ { x \rightarrow 6 ^ { - } } f ( x ) = - \infty , \lim _ { x \rightarrow 6 ^ { + } } f ( x ) = - \infty , \lim _ { x - 6 ^ { + } } f ( x ) = \infty , \lim _ { x \rightarrow 6 ^ { - } } f ( x ) = \infty$

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

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 $x , 0 < | \mathrm { x } - \mathrm { c } | < \delta \Rightarrow | \mathrm { f } ( \mathrm { x } ) - \mathrm { L } | < \varepsilon$ . - $f ( x ) = - 3 x - 7 , L = - 19 , c = 4$ , and $\varepsilon = 0.01$

Use the slopes of UQ, UR, US, and UT to estimate the rate of change of y at the specified value of x. - $x=2$

Divide numerator and denominator by the highest power of x in the denominator to find the limit. - $\lim _ { t \rightarrow \infty } \frac { \sqrt { 9 t ^ { 2 } - 27 } } { t - 3 }$

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit. -What conditions, when present, are sufficient to conclude that a function $f ( x )$ has a limit as $x$ approaches some value of a?

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

Find the limit. - $\lim _ { x \rightarrow 8 ^ { + } } \frac { \sqrt { 5 x } ( x - 3 ) } { | x - 3 | }$

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 $x , 0 < | \mathrm { x } - \mathrm { c } | < \delta \Rightarrow | \mathrm { f } ( \mathrm { x } ) - \mathrm { L } | < \varepsilon$ . - $f ( x ) = 7 x ^ { 2 } , L = 175 , c = 5 \text {, and } \varepsilon = 0.5$

Provide an appropriate response. -If $\lim _ { x \rightarrow \theta } f ( x ) = L$ , which of the following expressions are true? I. $\lim _ { x \rightarrow \theta ^ { - } } f ( x )$ does not exist. II. $\lim _ { x \rightarrow \theta ^ { + } } f ( x )$ does not exist. III. $\lim _ { x \rightarrow \theta ^ { - } } f ( x ) = L$ IV. $\lim _ { x \rightarrow \theta ^ { + } } f ( x ) = L$

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit. - $\lim _ { - 64 } \frac { 8 - \sqrt { x } } { 64 - x }$

Answer the question. -Does $\lim _ { x \rightarrow 2 } f ( x ) = f ( 2 )$ ? $f ( x ) = \left\{ \begin{array} { l l } x ^ { 3 } , & - 2 < x \leq 0 \\- 2 x , & 0 \leq x < 2 \\6 , & 2 < x \leq 4 \\0 , & x = 2\end{array} \right.$

Find the limit, if it exists. - $\lim _ { x \rightarrow 4 } \frac { x ^ { 2 } + 9 x + 20 } { x + 4 }$

Find numbers a and b, or k, so that f is continuous at every point. - $f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } , & \text { if } x \leq 6 \\x + k , & \text { if } x > 6\end{array} \right.$

Divide numerator and denominator by the highest power of x in the denominator to find the limit. - $\lim _{x \rightarrow\infty} \frac{5 x+3}{\sqrt{3 x^{2}+1}}$

Find the limit, if it exists. - $\lim _ { h \rightarrow 0 } \frac { ( 1 + h ) ^ { 1 / 3 } - 1 } { h }$

Find the limit. - $\lim _ { x \rightarrow 0.5 ^ { - } } \sqrt { \frac { x + 9 } { x + 2 } }$