91Ó°ÊÓ

Using the Definition of Limit The definition of limit on page 56 requires that \(f\) is a function defined on an open interval containing \(c\) , except possibly at \(c .\) Why is this requirement necessary?

Finding a Limit of a Trigonometric Function In Exercises \(63-74\) , find the limit of the trigonometric function. $$\lim _{x \rightarrow 0} \frac{3(1-\cos x)}{x}$$

Comparing Functions and Limits If\(f(2)=4\) can you conclude anything about the limit of \(f(x)\) as \(x\) approaches 2\(?\) Explain your reasoning.

The graphs of polynomial functions have no vertical asymptotes.

Comparing Functions and Limits If the limit of \(f(x)\) as \(x\) approaches 2 is \(4,\) can you conclude anything about \(f(2) ?\) Explain your reasoning.

If \(f\) has a vertical asymptote at \(x=0,\) then \(f\) is undefined at \(x=0 .\)

Continuity of a Composite Function In Exercises \(65-70\) , discuss the continuity of the composite function $$f(x)=\sin x$$ $$g(x)=x^{2}$$

True or False? In Exercises \(73-76,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is undefined at \(x=c\) , then the limit of \(f(x)\) as \(x\) approaches \(c\) does not exist.

Testing for Continuity In Exercises \(75-82,\) describe the interval(s) on which the function is continuous. $$f(x)=\frac{x+1}{\sqrt{x}}$$

Graphical, Numerical, and Analytic Analysis In Exercises \(75-82,\) use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. The linit. the limit by analytic methods $$\lim _{x \rightarrow 2} \frac{x^{3}-32}{x-2}$$