Exam 2: Limits and Derivatives

Use the table of values of f to estimate the limit. - $\text { Let } f(x)=\frac{x^{2}-4 x-5}{x^{2}-7 x+10} \text {, find } \lim _{x \rightarrow 5} f(x) \text {. }$ 4.9 4.99 4.999 5.001 5.01 5.1 ()

Find the limit. - $\lim _ { x \rightarrow 2 } \left( x ^ { 2 } + 8 x - 2 \right)$

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 $x _ { t } 0 < \left| x - x _ { 0 } \right| < \delta \Rightarrow | f ( x ) - L | < \varepsilon$ . - $f ( x ) = \sqrt { 18 - x } , L = 4 , x _ { 0 } = 2 , \text { and } \varepsilon = 1$

Use the table to estimate the rate of change of y at the specified value of x. -x = 1. 0 0 0.2 0.02 0.4 0.08 0.6 0.18 0.8 0.32 1.0 0.5 1.2 0.72 1.4 0.98

Use the graph to evaluate the limit. - $\lim _ { x \rightarrow 1 } f ( x )$

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 0 } \frac { \sin 5 x } { \sin 4 x }$

Use the slopes of UQ, UR, US, and UT to estimate the rate of change of y at the specified value of x. -x = 2.5

$\text { Find the limit using } \lim _ { x = 0 } \frac { \sin x } { x } = 1 \text {. }$ - $\lim _ { x \rightarrow 0 } \frac { \sin x \cos 4 x } { x + x \cos 5 x }$

Prove the limit statement -You are asked to make some circular cylinders, each with a cross-sectional area of $9 \mathrm {~cm} ^ { 2 }$ . To do this, you need to know how much deviation from the ideal cylinder diameter of $x _ { 0 } = 1.93 \mathrm {~cm}$ you can allow and still have the area come within $0.1 \mathrm {~cm} ^ { 2 }$ of the required $9 \mathrm {~cm} ^ { 2 }$ . To find out, let $A = \pi \left( \frac { x } { 2 } \right) ^ { 2 }$ and look for the interval in which you must hold $x$ to make $| \mathrm { A } - 9 | < 0.1$ . What interval do you find?

Find the limit if it exists. - $\lim _ { x \rightarrow 1 } 9 x ( x + 4 ) ( x - 4 )$

Use the graph to estimate the specified limit. - $\text { Find } \lim _ { x \rightarrow 0 } f ( x )$

Provide an appropriate response. -Given $\varepsilon > 0$ , find an interval $I = ( 5 - \delta , 5 ) , \delta > 0$ , such that if $x$ lies in $I$ , then $\sqrt { 5 - x } < \varepsilon$ . What limit is being verified and what is its value?

Provide an appropriate response. -Write the formal notation for the principle "the limit of a quotient is the quotient of the limits" and include a statement of any restrictions on the principle.

The graph below shows the number of tuberculosis deaths in the United States from 1989 to 1998. Estimate the average rate of change in tuberculosis deaths from 1993 to 1995.

Find all points where the function is discontinuous. -

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

Use the graph to estimate the specified limit. -Find lim x $\rightarrow$ 0 f(x)

Use the table of values of f to estimate the limit. - Let f(x)=, find f(x) x -0.1 -0.01 -0.001 0.001 0.01 0.1 f(x) 5.99640065 5.99640065

Prove the limit statement -Select the correct statement for the definition of the limit: $\lim _ { x \rightarrow 0 } f ( x ) = L$ means that___________

Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit. - $\lim _ { x \rightarrow 0 } \frac { \sqrt { 1 - x } - 1 } { x }$