91Ó°ÊÓ

Each of the functions in Problems 5 through 10 is either continuous on \((-\infty, \infty)\) or has a point of discontinuity at some point \((s) x=a .\) Determine any point \((s)\) of discontinuity. Is the point of discontinuity removable? In other words, can the function be made continuous by defining or redefining the function at the point of discontinuity? $$ f(x)=\frac{1}{x+2} $$

\(f(x)=|x-2| ;\) (a) \(\lim _{x \rightarrow 0} f(x)\) (b) \(\lim _{x \rightarrow 2} f(x)\)

Each of the functions in Problems 5 through 10 is either continuous on \((-\infty, \infty)\) or has a point of discontinuity at some point \((s) x=a .\) Determine any point \((s)\) of discontinuity. Is the point of discontinuity removable? In other words, can the function be made continuous by defining or redefining the function at the point of discontinuity? $$ f(x)=\frac{1}{x^{2}+2} $$

Let \(f\) be the function de ned by \(f(x)=\left\\{\begin{array}{ll}x+1, & \text { for } x \text { not an integer, } \\ 0, & \text { for } x \text { an integer. }\end{array}\right.\) (a) Sketch \(f\). (b) Find the following limits. i. \(\lim _{x \rightarrow 1.5} f(x)\) ii. \(\lim _{x \rightarrow 2} f(x)\) iii. \(\lim _{x \rightarrow 0} f(x)\) (c) For what values of \(c\) is \(\lim _{x \rightarrow c} f(x)=c+1 ?\) Have you excluded any values of \(c\) ? If so, which ones and why? Explain.

\(f(x)=\frac{x^{2}-2 x}{x-2}\) (a) \(\lim _{x \rightarrow 0} f(x)\) (b) \(\lim _{x \rightarrow 2} f(x)\)

\(f(x)=\left\\{\begin{array}{ll}5 x+1, & x \neq 2 \\ 7, & x=2\end{array}\right.\) (a) \(\lim _{x \rightarrow 1} f(x)\) (b) \(\lim _{x \rightarrow 2} f(x)\)

Each of the functions in Problems 5 through 10 is either continuous on \((-\infty, \infty)\) or has a point of discontinuity at some point \((s) x=a .\) Determine any point \((s)\) of discontinuity. Is the point of discontinuity removable? In other words, can the function be made continuous by defining or redefining the function at the point of discontinuity? $$ f(x)=\left\\{\begin{array}{ll} -x^{2}-1, & x>0 \\ 5 x-1, & x<0 \end{array}\right. $$

Let \(f\) be the function given by $$ f(x)=\left\\{\begin{array}{ll} \frac{1}{x+7}, & \text { for } x \leq-5 \\ x+4, & \text { for }-52 \end{array}\right. $$ Evaluate the limits below. (a) \(\lim _{x \rightarrow-\infty} f(x)\) (b) \(\lim _{x \rightarrow-7} f(x)\) (c) \(\lim _{x \rightarrow-5^{+}} f(x)\) (d) \(\lim _{x \rightarrow-5} f(x)\) (e) \(\lim _{x \rightarrow-2} f(x)\) (f) \(\lim _{x \rightarrow 2^{+}} f(x)\) (g) \(\lim _{x \rightarrow \infty} f(x)\) (h) \(\lim _{x \rightarrow 0} f(x)\)

\(f(x)=\frac{(x+2)\left(x^{2}-x\right)}{x(x-1)}\) (a) \(\lim _{x \rightarrow 0} f(x)\) (b) \(\lim _{x \rightarrow 1} f(x)\)

Each of the functions in Problems 5 through 10 is either continuous on \((-\infty, \infty)\) or has a point of discontinuity at some point \((s) x=a .\) Determine any point \((s)\) of discontinuity. Is the point of discontinuity removable? In other words, can the function be made continuous by defining or redefining the function at the point of discontinuity? $$ f(x)=\left\\{\begin{array}{ll} -x^{2}-x, & x>0 \\ 5 x-1, & x<0 \end{array}\right. $$