Exam 2: Limits and Derivatives

At what value(s) of x is the function $f ( x ) = \left\{ \begin{array} { l l } | x + 1 | - 1 & \text { if } x < 0 \\x ^ { 2 } + x & \text { if } 0 \leq x < 1 \\3 - x & \text { if } 1 \leq x\end{array} \right.$ discontinuous?

Find the value of the limit $\lim _ { x \rightarrow 1 } \frac { x - 1 } { \sqrt { x } - 1 }$

At what value(s) of x does the function $\frac { ( x + 2 ) ^ { 2 } } { x ^ { 2 } - 4 }$ have an infinite discontinuity?

Use a table of values to estimate the value of each of the following limits, to 4 decimal places.(a) $\lim _ { x \rightarrow 0 } \frac { 3 ^ { x } - 1 } { x }$ (b) $\lim _ { x \rightarrow 0 } \frac { \sin 5 x } { \sin 3 x }$ (c) $\lim _ { x \rightarrow 0 } ( 1 + x ) ^ { 1 / x }$

A car on a test track accelerates from 0 ft/s to 208 ft/s in 8 seconds. The car’s velocity is given in the table below: t(s) 0 1 2 3 4 5 6 7 8 v(t)(ft/s) 0 18 47 77 104 132 163 184 208 -On what time interval does the car have the greatest average acceleration?

Find the value of the limit $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 3 } - 1 } .$

Given the following information about limits, sketch a graph which could be the graph of y = f (x). Label all horizontal and vertical asymptotes. $\lim _ { x \rightarrow \infty } f ( x ) = \lim _ { x \rightarrow - \infty } f ( x ) = 0 , \quad \lim _ { x \rightarrow 1 } f ( x ) = \infty \lim _ { x \rightarrow - 1 } f ( x ) = - \infty , \text { and } f ( 0 ) = 0$