91Ó°ÊÓ

Find \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}\). $$ f(x)=2 x+3 $$

Use the properties of logarithms to expand the logarithmic expression. $$ \ln \frac{2}{3} $$

Describe the interval(s) on which the function is continuous. $$ f(x)=\sec \frac{\pi x}{4} $$

Jewelry A jeweler resizes a ring so that its inner circumference is 6 centimeters. (a) What is the radius of the ring? (b) If the ring's inner circumference can vary between 5.5 centimeters and 6.5 centimeters, how can the radius vary? (c) Use the \(\varepsilon-\delta\) definition of a limit to describe this situation. Identify \(\varepsilon\) and \(\delta\).

True or False? In Exercises \(50-53\), determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) has a vertical asymptote at \(x=0,\) then \(f\) is undefined at \(x=0\)

Show that \(f\) is one-to-one on the indicated interval and therefore has an inverse function on that interval. $$ f(x)=\cos x \quad[0, \pi] $$

$$ \begin{aligned} &\text { Show that the distance between the point }\left(x_{1}, y_{1}\right) \text { and the line }\\\ &A x+B y+C=0 \text { is }\\\ &\text { Distance }=\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}} \text { . } \end{aligned} $$

Describe the interval(s) on which the function is continuous. $$ f(x)=\frac{x+1}{\sqrt{x}} $$

Write the distance \(d\) between the point (3,1) and the line \(y=m x+4\) in terms of \(m .\) Use a graphing utility to graph the equation. When is the distance 0 ? Explain the result geometrically.

Show that \(f\) is one-to-one on the indicated interval and therefore has an inverse function on that interval. $$ f(x)=\cot x \quad (0, \pi) $$