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Chapter 1: Problem 24
Find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? $$ f(x)=\frac{1}{x^{2}+1} $$
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In Exercises \(35-38\), use a graphing utility to graph the function and determine the one-sided limit. $$ \begin{array}{l} f(x)=\frac{x^{3}-1}{x^{2}+x+1} \\ \lim _{x \rightarrow 1^{-}} f(x) \end{array} $$
In Exercises \(35-38\), use a graphing utility to graph the function and determine the one-sided limit. $$ \begin{array}{l} f(x)=\frac{1}{x^{2}-25} \\ \lim _{x \rightarrow 5^{-}} f(x) \end{array} $$
Write the expression in algebraic form. \(\sin (\operatorname{arcsec} x)\)
Show that the Dirichlet function \(f(x)=\left\\{\begin{array}{ll}0, & \text { if } x \text { is rational } \\\ 1, & \text { if } x \text { is irrational }\end{array}\right.\) is not continuous at any real number.
Numerical and Graphical Analysis Use a graphing utility to complete the table for each function and graph each function to estimate the limit. What is the value of the limit when the power on \(x\) in the denominator is greater than \(3 ?\) $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 1 & 0.5 & 0.2 & 0.1 & 0.01 & 0.001 & 0.0001 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & & \\ \hline \end{array} $$ (a) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x}\) (b) \(\lim _{x \rightarrow 0^{-}} \frac{x-\sin x}{x^{2}}\) (c) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{3}}\) (d) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{4}}\)
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