Exam 3: Limits and Continuity

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 $f$ . - $f ( x ) = 2 \sqrt { x } + 10 , x = 25$

Find the limit. - $\lim _ { x \rightarrow - 0 } \frac { 1 } { x ^ { 2 / 3 } }$

Find numbers a and b, or k, so that f is continuous at every point. - $f ( x ) = \left\{ \begin{array} { l l } 10 x + 8 , & \text { if } x < - 7 \\ k x + 2 , & \text { if } x \geq - 7 \end{array} \right.$

Find the limit. - $\lim _ { x \rightarrow \infty } \frac { \cos 5 x } { x }$

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

Find the limit. - $\lim _ { x \rightarrow } \left( 4 x - \sqrt { 16 x ^ { 2 } - 3 x + 6 } \right)$

Find the intervals on which the function is continuous. - $y = \frac { \sin ( 3 \theta ) } { 5 \theta }$

Find all points where the function is discontinuous. -

Find all points where the function is discontinuous. -

Find the limit if it exists. - $m_{-14} \sqrt{5}$

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

Find the limit. - $\lim _ { x \rightarrow \infty } \frac { 4 x ^ { 3 } + 3 x ^ { 2 } } { x - 6 x ^ { 2 } }$

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

Provide an appropriate response. -Explain why the following five statements ask for the same information. (a) Find the roots of $f ( x ) = 4 x ^ { 3 } - 3 x - 4$ . (b) Find the $x$ -coordinate of the points where the curve $y = 4 x ^ { 3 }$ crosses the line $y = 3 x + 4$ . (c) Find all the values of $x$ for which $4 x ^ { 3 } - 3 x = 4$ . (d) Find the $x$ -coordinates of the points where the cubic curve $y = 4 x ^ { 3 } - 3 x$ crosses the line $y = 4$ . (e) Solve the equation $4 x ^ { 3 } - 3 x - 4 = 0$ .

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 $x , 0 < | \mathrm { x } - \mathrm { c } | < \delta \Rightarrow | \mathrm { f } ( \mathrm { x } ) - \mathrm { L } | < \varepsilon$ . - $f ( x ) = m x + b , m > 0 , L = \frac { m } { 5 } + b , c = \frac { 1 } { 5 } , \text { and } \varepsilon = c > 0$

Find the limit. - $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x } = 1$ , find $\lim _ { x \rightarrow \theta } f ( x )$

Find the limit. - $\lim _ { x \rightarrow \pi } \sqrt { x + 7 } \cos ( x + \pi )$

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

Solve the problem. -Ohm's Law for electrical circuits is stated $\mathrm { V } = \mathrm { RI }$ , where $\mathrm { V }$ is a constant voltage, $\mathrm { R }$ is the resistance in ohms and I is the current in amperes. Your firm has been asked to supply the resistors for a circuit in which $\mathrm { V }$ will be 12 volts and I is to be $5 \pm 0.1$ amperes. In what interval does $R$ have to lie for I to be within $0.1$ amps of the target value $I _ { 0 } = 5$ ?

Find the intervals on which the function is continuous. - $y = \sqrt [ 4 ] { 2 x - 4 }$