91Ó°ÊÓ

Find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? $$ f(x)=\left\\{\begin{array}{ll} -2 x+3, & x<1 \\ x^{2}, & x \geq 1 \end{array}\right. $$

Find the limit of the function (if it exists). Write a simpler function that agrees with the given function at all but one point. Use a graphing utility to confirm your result. $$ \lim _{x \rightarrow 2} \frac{x^{3}-8}{x-2} $$

Find the limit \(L\). Then use the \(\varepsilon-\delta\) definition to prove that the limit is \(L\). $$ \lim _{x \rightarrow 1}\left(x^{2}+1\right) $$

Find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? $$ f(x)=\left\\{\begin{array}{ll} \frac{1}{2} x+1, & x \leq 2 \\ 3-x, & x>2 \end{array}\right. $$

Find the limit. \(\lim _{x \rightarrow 1 / 2} x \sec \pi x\)

Find the limit of the function (if it exists). Write a simpler function that agrees with the given function at all but one point. Use a graphing utility to confirm your result. $$ \lim _{x \rightarrow-1} \frac{x^{3}+1}{x+1} $$

Find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? $$ f(x)=\left\\{\begin{array}{ll} -2 x, & x \leq 2 \\ x^{2}-4 x+1, & x>2 \end{array}\right. $$

Find the limit. \(\lim _{x \rightarrow 1 / 2} x^{2} \tan \pi x\)

Find the limit \(L\). Then use the \(\varepsilon-\delta\) definition to prove that the limit is \(L\). $$ \lim _{x \rightarrow-3}\left(x^{2}+3 x\right) $$

Find the limit (if it exists). $$ \lim _{x \rightarrow 5} \frac{x-5}{x^{2}-25} $$