Exam 3: Limits and Continuity

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." - $f ( x ) = \frac { 10 ^ { 2 x } - 1 } { x }$

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

Find the limit if it exists. - $\lim _ { x \rightarrow \frac { 2 } { 5 } } 5 x \left( x - \frac { 1 } { 3 } \right)$

Find the average rate of change of the function over the given interval. - $y = \frac { 3 } { x - 2 } , [ 4,7 ]$

Find the limit and determine if the function is continuous at the point being approached. - $\lim _ { x \rightarrow - \pi } \sin ( 2 x - \sin 2 x )$

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit. - $\lim _ { x \rightarrow 0 } \frac { \sqrt { 64 - x } - 8 } { x }$

Provide an appropriate response. -Identify the incorrect statements about limits. I. The number $L$ is the limit of $f ( x )$ as $x$ approaches $c$ if $f ( x )$ gets closer to $L$ as $x$ approaches $x _ { 0 }$ . II. The number $L$ is the limit of $f ( x )$ as $x$ approaches $c$ if, for any $\varepsilon > 0$ , there corresponds a $\delta > 0$ such that $\mid f ( x ) - L$ $\varepsilon$ whenever $0 < | x - c | < \delta$ . III. The number $L$ is the limit of $f ( x )$ as $x$ approaches $c$ if, given any $\varepsilon > 0$ , there exists a value of $x$ for which $\mid f ( x )$ $\varepsilon .$