Exam 3: Limits and Continuity

For the function f whose graph is given, determine the limit. - $\text { Find } \lim _ { x \rightarrow 2 ^ { - } } f ( x ) \text { and } \lim _ { x - 2 ^ { + } } f ( x ) \text {. }$

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

Find the limit. - $\lim _ { x \rightarrow 2 ^ { + } } \sqrt { \frac { 7 x ^ { 2 } } { 10 + x } }$

Find the limit. - $\lim _ { x \rightarrow \infty } \frac { - 7 x ^ { 2 } - 9 x + 2 } { - 13 x ^ { 2 } - 8 x + 3 }$

Find the limit. - $\lim _ { x \rightarrow 2 ^ { + } } \left( \frac { x } { x + 2 } \right) \left( \frac { - 4 x + 4 } { x ^ { 2 } + 2 x } \right)$

Find the limit. - $\lim _ { x \rightarrow 0 } \left( x ^ { 2 } - 5 \right)$

Find the limit. - $\lim _{x \rightarrow-\pi / 2)^{-}} \sec x$

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 $x , 0 < | \mathrm { x } - \mathrm { c } | < \delta \Rightarrow | \mathrm { f } ( \mathrm { x } ) - \mathrm { L } | < \varepsilon$ . - $\mathrm { f } ( \mathrm { x } ) = \frac { 1 } { \mathrm { x } } , \mathrm { L } = \frac { 1 } { 2 } , \mathrm { c } = 2$ , and $\varepsilon = 0.3$

Given the interval $( a , b )$ on the $x$ -axis with the point $c$ inside, find the greatest value for $\delta > 0$ such that for all $x$ , $0 < | x - c | < \delta \Rightarrow a < x < b \text {. }$ - $a = 1.348 , b = 2.804 , c = 1.869$

Use the table of values of f to estimate the limit. -Let $f ( x ) = \frac { \sin ( 3 x ) } { x }$ , find $\lim _ { x \rightarrow - 0 } f ( x )$ x -0.1 -0.01 -0.001 0.001 0.01 0.1 () 2.99955002 2.99955002

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 $x , 0 < | \mathrm { x } - \mathrm { c } | < \delta \Rightarrow | \mathrm { f } ( \mathrm { x } ) - \mathrm { L } | < \varepsilon$ . - $f ( x ) = \sqrt { 58 - 3 x } , c = - 2 , \varepsilon = 0.2$

Use the graph to evaluate the limit. - $\lim _ { x \rightarrow - 0 } f ( x )$

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit. - $\lim _ { x \rightarrow 6 } \frac { \sqrt { x } - 4 } { x - 16 }$

Find the limit, if it exists. - $\lim _ { x \rightarrow 1 } \frac { 2 x - 7 } { 4 x + 5 }$

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 $x , 0 < | \mathrm { x } - \mathrm { c } | < \delta \Rightarrow | \mathrm { f } ( \mathrm { x } ) - \mathrm { L } | < \varepsilon$ . - $\mathrm { f } ( \mathrm { x } ) = \sqrt { 17 - \mathrm { x } } , \mathrm { L } = 4 , \mathrm { c } = 1$ , and $\varepsilon = 1$

Find the limit. - $\lim _ { x \rightarrow \sqrt [ 3 ] { 5 } } \frac { x ^ { 2 } } { 5 } - \frac { 1 } { x }$

Graph the rational function. Include the graphs and equations of the asymptotes. - $y = \frac { 1 } { ( x + 2 ) ^ { 2 } }$

Provide an appropriate response. -Let $\lim _ { x \rightarrow 2 } f ( x ) = - 6$ and $\lim _ { x \rightarrow 2 } g ( x ) = - 8$ . Find $\lim _ { x \rightarrow 2 } [ f ( x ) \cdot g ( x ) ]$ .

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 $f$ . - $f ( x ) = 5 x ^ { 2 } , x = - 2$

Provide an appropriate response. -Given $\varepsilon > 0$ , find an interval $\mathrm { I } = ( 3 - \delta , 3 ) , \delta > 0$ , such that if $x$ lies in $\mathrm { I }$ , then $\sqrt { 3 - x } < \varepsilon$ . What limit is being verified and what is its value?