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 $x _ { t } 0 < \left| x - x _ { 0 } \right| < \delta \Rightarrow | f ( x ) - L | < \varepsilon$ . - $f ( x ) = \frac { x ^ { 2 } + 4 x + - 21 } { x + 7 } , x _ { 0 } = - 7 , \varepsilon = 0.03$

Use the table to estimate the rate of change of y at the specified value of x. -When exposed to ethylene gas, green bananas will ripen at an accelerated rate. The number of days for ripening becomes shorter for longer exposure times. Assume that the table below gives average ripening times of Bananas for several different ethylene exposure times: Exposure timeRipening Time (minutes) (days) 10 4.2 15 3.5 20 2.6 25 2.1 30 1.1 Plot the data and then find a line approximating the data. With the aid of this line, find the limit of the average ripening Time as the exposure time to ethylene approaches 0. Round your answer to the nearest tenth.

Provide an appropriate response. -The statement "the limit of a constant times a function is the constant times the limit" follows from a combination of two fundamental limit principles. What are they?

Provide an appropriate response. -Given $\lim _ { x \rightarrow 0 ^ { - } } \mathrm { f } ( \mathrm { x } ) = \mathrm { L } _ { \mathrm { l } } , \lim _ { \mathrm { x } - 0 ^ { + } } \mathrm { f } ( \mathrm { x } ) = \mathrm { L } _ { \mathrm { r } }$ , and $\mathrm { L } \mathrm { l } \neq \mathrm { L } _ { \mathrm { r } }$ , which of the following statements is true? I. $\lim _ { x \rightarrow 0 } f ( x ) = L _ { l }$ II. $\lim _ { x \rightarrow 0 } f ( x ) = L _ { r }$ III. $\lim _ { x \rightarrow 0 } f ( x )$ does not exist.

Find the limit. - $\lim _ { x \rightarrow 1 } \frac { x } { 3 x + 2 }$

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 $x _ { t } 0 < \left| x - x _ { 0 } \right| < \delta \Rightarrow | f ( x ) - L | < \varepsilon$ . - $f ( x ) = \sqrt { x + 5 } , L = 3 , x 0 = 4 , \text { and } \varepsilon = 1$

Give an appropriate answer. - $f ( x ) = 3 x ^ { 2 } + 4$ for $x _ { 0 } = 4$

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

Find the limit if it exists. - $\lim _ { x \rightarrow 8 } \left( 5 x ^ { 2 } - 3 x - 6 \right)$

Find the limit, if it exists. - $\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } - 49 } { x - 7 }$

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

Find the limit, if it exists. - $\lim _ { h \rightarrow 0 } \frac { 7 x + h } { x ^ { 3 } ( x - h ) }$

Find all points where the function is discontinuous. -

Find the limit if it exists. - $\lim _ { x - 625 } x ^ { 3 / 4 }$

Use the table of values of f to estimate the limit. - $\text { Let } f(x)=x^{2}-5 \text {, find } \lim _{x \rightarrow \theta} f(x)$ -0.1 -0.01 -0.001 0.001 0.01 0.1 ()

Find all points where the function is discontinuous. -

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 $x _ { t } 0 < \left| x - x _ { 0 } \right| < \delta \Rightarrow | f ( x ) - L | < \varepsilon$ . -

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit. - $\lim _ { x \rightarrow 0 } \frac { 4 - \sqrt { 16 - x ^ { 2 } } } { x }$

Find the limit, if it exists. - $\lim _ { x \rightarrow 6 } \frac { x ^ { 2 } + 3 x - 18 } { x ^ { 2 } + 2 x - 24 }$

Find the limit. -If $\lim _ { x \rightarrow 1 } \frac { f ( x ) - 3 } { x - 4 } = 4$ , find $\lim _ { x \rightarrow 1 } f ( x )$ .