Exam 13: Limits: a Preview of Calculus

Find the derivative of the function at the given number. $f ( x ) = 2 - x ^ { 2 }$ at 2

Use the definition of area as a limit to find the area of the region that lies under the graph of $f ( x ) = 9 - x ^ { 2 }$ over the interval $- 3 \leq x \leq 3$ .

Find the slope of the tangent line of the graph of $f ( x ) = x - 3 x ^ { 2 }$ at the point $( - 1 , - 4 )$ .

Estimate the area under the graph of $f ( x ) = 2 x + 1$ from $x = 0$ to $x = 4$ using (a) four approximating rectangles and right endpoints. (b) four approximating rectangles and left endpoints. (c) eight approximating rectangles and right endpoints.

Use a table of values to estimate the value of $\lim _ { x \rightarrow \infty } \frac { x } { \sqrt { 2 x ^ { 2 } + 1 } }$ . Then use a graphing device to confirm your result graphically.

Find the derivative of the function at the given number. $f ( x ) = x ^ { 2 } - x ^ { 3 }$ at 1

Find the derivative of the function at the given number. $g ( x ) = 1 + \sqrt { x }$ at 4

Use a table of values to estimate the value of $\lim _ { x \rightarrow 0 } \frac { 6 - 3 \sqrt { 4 + x } } { x }$ . Use a graphing device to confirm your answer graphically.

Evaluate $\lim _ { x \rightarrow 0 } \frac { 2 x ^ { 2 } + x - 9 } { x - 2 }$ , and justify each step by indicating the appropriate Limit Law(s).

Determine whether the sequence $a _ { n } = \frac { ( - 1 ) ^ { n - 1 } } { n . 1 }$ converges or diverges. If it converges, find the limit.

Find the derivative of the function at the given number. $g ( x ) = x ^ { 2 } + x ^ { 3 }$ at 1

Evaluate the limit if it exists. $\lim _ { h \rightarrow 0 } \frac { ( 4 + h ) ^ { - 1 } - 4 ^ { - 1 } } { h }$

Determine whether the sequence $a _ { n } = \lim _ { n \rightarrow \infty } \frac { 5 } { 4 n ^ { 4 } } \left[ \frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 4 } \right]$ converges or diverges. If it converges, find the limit.

Find $f ^ { \prime } ( a )$ , where a is in the domain of f. $f ( x ) = \sqrt { x - 4 }$

Evaluate $\lim _ { t \rightarrow 8 } \frac { 4 - \sqrt { t + 8 } } { 8 - t }$ .

Evaluate the limit if it exists. $\lim _ { t \rightarrow 0 } \left( \frac { 1 } { 2 t } - \frac { 1 } { t ^ { 2 } + 2 t } \right)$

Estimate the area under the graph of $f ( x ) = 2 ^ { - x }$ from $x = 0$ to $x = 4$ using (a) four approximating rectangles and right endpoints. (b) four approximating rectangles and left endpoints. (c) eight approximating rectangles and right endpoints.

Find the value of $\lim _ { x \rightarrow 1 } \frac { 2 x - x ^ { 2 } - 1 } { | x - 1 | }$ , if it exists. If the limit does not exist, explain why.

Use a table of values to estimate the value of $\lim _ { x \rightarrow \infty } \frac { x ^ { 6 } } { 2 ^ { x } }$ . Then use a graphing device to confirm your result graphically.

Find the value of $\lim _ { x \rightarrow 1 } \frac { | 1 - x | } { 1 - x }$ , if it exists. If the limit does not exist, explain why.