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Chapter 1: Problem 5
Find the limit. $$ \lim _{x \rightarrow-3}\left(2 x^{2}+4 x+1\right) $$
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(a) Prove that if \(\lim _{x \rightarrow c}|f(x)|=0,\) then \(\lim _{x \rightarrow c} f(x)=0\). (Note: This is the converse of Exercise \(74 .)\) (b) Prove that if \(\lim _{x \rightarrow c} f(x)=L,\) then \(\lim _{x \rightarrow c}|f(x)|=|L|\). [Hint: Use the inequality \(\|f(x)|-| L\| \leq|f(x)-L| .]\)
Use a graphing utility to graph the function on the interval \([-4,4] .\) Does the graph of the function appear continuous on this interval? Is the function continuous on [-4,4]\(?\) Write a short paragraph about the importance of examining a function analytically as well as graphically. $$ f(x)=\frac{e^{-x}+1}{e^{x}-1} $$
In Exercises \(35-38\), use a graphing utility to graph the function and determine the one-sided limit. $$ \begin{array}{l} f(x)=\frac{x^{3}-1}{x^{2}+x+1} \\ \lim _{x \rightarrow 1^{-}} f(x) \end{array} $$
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. The graphs of polynomial functions have no vertical asymptotes.
Use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval [0, 1]. Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to approximate the zero accurate to four decimal places. $$ h(\theta)=1+\theta-3 \tan \theta $$
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