Exam 3: Limits and Continuity

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

Find the limit if it exists. - $\lim _ { x \rightarrow 7 } ( 3 - 10 x )$

For the function f whose graph is given, determine the limit. -Find $\lim _ { x \rightarrow 7 ^ { - } } f ( x )$

Use the graph to estimate the specified limit. -Find $\lim _ { x \rightarrow \pi / 2 ) ^ { - } } f ( x )$ and $\lim _ { x \rightarrow \pi / 2 ) ^ { + } } f ( x )$

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

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

Find the limit. - $\lim _ { x \rightarrow 7 ^ { + } } \frac { 1 } { ( x - 7 ) ^ { 2 } }$

Provide an appropriate response. -A function $\mathrm { y } = \mathrm { f } ( \mathrm { x } )$ is continuous on $[ - 1,1 ]$ . It is known to be positive at $x = - 1$ and negative at $x = 1$ . What, if anything, does this indicate about the equation $f ( x ) = 0$ ? Illustrate with a sketch.

Provide an appropriate response. - $\text { Let } \lim _ { x \rightarrow 3 } f ( x ) = - 5 \text { and } \lim _ { x \rightarrow 3 } g ( x ) = - 7 \text {. Find } \lim _ { x \rightarrow 3 } [ f ( x ) + g ( x ) ] ^ { 2 } \text {. }$

Find the limit. - $\lim _ { x \rightarrow 2 } \sqrt { x ^ { 2 } + 2 x + 1 }$

Provide an appropriate response. -If functions $f ( x )$ and $g ( x )$ are continuous for $0 \leq x \leq 6 , \operatorname { could } \frac { f ( x ) } { g ( x ) }$ possibly be discontinuous at a point of $[ 0,6 ] ?$ Provide an example.

Find the limit. - $\lim _{x-(\pi / 2)^{+}} \tan x$

Use the table to estimate the rate of change of y at the specified value of x. - $x= 1 .$ x y 0 0 2 0.01 4 0.04 6 0.09 8 0.16 0 0.25 2 0.36 4 0.49

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

Provide an appropriate response. - $\text { Use the Intermediate Value Theorem to prove that } 7 \sin x = x \text { has a solution between } \frac { \pi } { 2 } \text { and } \pi \text {. }$

Find the slope of the curve at the given point P and an equation of the tangent line at P. - $y = x ^ { 2 } + 11 x - 15 , P ( 1 , - 3 )$

Sketch the graph of a function y = f(x) that satisfies the given conditions. - $\lim _ { x \rightarrow \infty } f ( x ) = 0 , \lim _ { x \rightarrow 2 ^ { - } } f ( x ) = \infty , \lim _ { x \rightarrow 2 ^ { + } } f ( x ) = \infty ,$

Find the limit, if it exists. - $\lim _ { x \rightarrow - } \frac { x ^ { 3 } + 12 x ^ { 2 } - 5 x } { 5 x }$

Find the limit, if it exists. - $\lim _ { x \rightarrow 4 } \frac { x ^ { 2 } - 16 } { x - 4 }$

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