Exam 2: Limits and Derivatives

Find the limit $L$ for the given function $f$ , the point $x _ { 0 }$ , and the positive number $\varepsilon$ . Then find a number $\delta > 0$ such that, for all $x _ { t } 0 < \left| x - x _ { 0 } \right| < \delta \Rightarrow | f ( x ) - L | < \varepsilon$ . - $f ( x ) = 1 / x , L = 1 / 7 , x _ { 0 } = 7 \text {, and } \varepsilon = 0.4$

Provide an appropriate response. -Provide a short sentence that summarizes the general limit principle given by the formal notation $\lim _ { x \rightarrow a } [ f ( x ) \pm g ( x ) ] = \lim _ { x \rightarrow a } f ( x ) \pm \lim _ { x \rightarrow a } g ( x ) = L \pm M$ , given that $\lim _ { x \rightarrow a } f ( x ) = L$ and $\lim _ { x \rightarrow a } g ( x ) = M$ .

Provide an appropriate response. -If $\lim _ { x \rightarrow 1 ^ { - } } f ( x ) = 1$ , $\lim _ { x \rightarrow 1 ^ { + } } f ( x ) = - 1$ , and $f ( x )$ is an even function, which of the following statements are true? I. $\lim _ { x \rightarrow 1 ^ { - } } f ( x ) = - 1$ II. $\lim _ { x \rightarrow 1 ^ { + } } f ( x ) = - 1$ III. $\lim _ { x \rightarrow 1 } f ( x )$ does not exist.

Find all points where the function is discontinuous. -

Find the limit, if it exists. - $\lim _ { x \rightarrow 1 } \frac { 3 x ^ { 2 } + 7 x - 2 } { 3 x ^ { 2 } - 4 x - 2 }$

Find the limit, if it exists. - $\lim _ { x \rightarrow 8 } \frac { x ^ { 2 } + 2 x - 80 } { x ^ { 2 } - 64 }$

Find the average rate of change of the function over the given interval. - $y = \sqrt { 2 x } , [ 2,8 ]$

Find the limit, if it exists. - $\lim _ { x \rightarrow 14 } \frac { 1 } { x - 14 }$

Find the limit if it exists. - $\lim _ { x \rightarrow 9 } ( 22 - 6 x )$

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 2 x + \sin x } { x }$

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit. - $\lim _ { x \rightarrow- 1 } \frac { x ^ { 2 } - 1 } { \sqrt { x ^ { 2 } + 3 } - 2 }$

Provide an appropriate response. -It can be shown that the inequalities $- x \leq x \cos \left( \frac { 1 } { x } \right) \leq x$ hold for all values of $x \geq 0$ . Find $\lim _ { x \rightarrow 0 } x \cos \left( \frac { 1 } { x } \right)$ if it exists.

Provide an appropriate response. -If $\lim _ { x \rightarrow 0 } f ( x ) = L$ , which of the following expressions are true? I. $\lim _ { x \rightarrow 0 ^ { - } } f ( x )$ does not exist. II. $\lim _ { x \rightarrow0 ^ { + } } f ( x )$ does not exist. III. $\lim _ { x \rightarrow0 ^ { - } } f ( x ) = L$ IV. $\lim _ { x \rightarrow0 ^ { + } } f ( x ) = L$

Find the limit $L$ for the given function $f$ , the point $x _ { 0 }$ , and the positive number $\varepsilon$ . Then find a number $\delta > 0$ such that, for all $x _ { t } 0 < \left| x - x _ { 0 } \right| < \delta \Rightarrow | f ( x ) - L | < \varepsilon$ . - $f ( x ) = m x + b , m > 0 , L = ( m / 8 ) + b , x _ { 0 } = 1 / 8 \text {, and } \varepsilon = c > 0$

Find the limit $L$ for the given function $f$ , the point $x _ { 0 }$ , and the positive number $\varepsilon$ . Then find a number $\delta > 0$ such that, for all $x _ { t } 0 < \left| x - x _ { 0 } \right| < \delta \Rightarrow | f ( x ) - L | < \varepsilon$ . -

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow0 } \frac { \tan 4 x } { x }$

Find all points where the function is discontinuous. -

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit. - $\lim _ { x \rightarrow 0 } \frac { \sqrt { 81 + x } - \sqrt { 81 - x } } { x }$

Give an appropriate answer. - $f ( x ) = 4 x + 5 \text { for } x _ { 0 } = 2$

Find the limit $L$ for the given function $f$ , the point $x _ { 0 }$ , and the positive number $\varepsilon$ . Then find a number $\delta > 0$ such that, for all $x _ { t } 0 < \left| x - x _ { 0 } \right| < \delta \Rightarrow | f ( x ) - L | < \varepsilon$ . - $f ( x ) = m x , m > 0 , L = 4 m , x _ { 0 } = 4$ , and $\varepsilon = 0.07$