91Ó°ÊÓ

The function \(f\) is not defined at \(x=0\). Use a graphing utility to graph \(f\). Zoom in to determine whether there is a number \(k\) such that the function $$F(x) \quad\left\\{\begin{aligned} f(x), & x \neq 0 \\ k, & x=0 \end{aligned}\right.$$ is continuous at \(x=0\). If so, what is \(k ?\) Support your conclusion by calculating the limit using a CAS. $$f(x)=\frac{x \sin 2 x}{\sin x^{2}}$$.

Use a graphing utility to find at least one number \(c\) at which $$\lim _{x \rightarrow 6} f(x)$$ does not exist. $$f(x)=\frac{\left|6 x^{2}-x-35\right|}{2 x-5}$$

Prove or give a counterexample: if \(f(c)>0\) and \(\lim _{x \rightarrow c} f(x)\) exists, then \(f(x)>0\) for all \(x\) in an interval of the form \((c-y, c+\gamma)\)

Suppose that \(f(x) \leq g(x)\) for all \(x \in(c-p, c+p),\) except possibly at \(c\) itself. (a) Prove that \(\lim _{x \rightarrow c} f(x) \leq \lim _{x \rightarrow c} g(x),\) provided cach of these limits exist. (b) Suppose that \(f(x)

Use a graphing utility to draw the graphs of $$f(x)=\frac{1}{x} \sin x$$ and $$g(x)=x \sin \left(\frac{1}{x}\right)$$ for \(x \neq 0\) between \(-\pi / 2\) and \(\pi / 2\). Describe tie behavior of \(f(x)\) and \(g(x)\) for \(x\) close to 0

Use a graphing utility to draw the graphs of $$f(x)=\frac{1}{x} \tan x$$ and $$g(x)=\ x tan \left(\frac{1}{x}\right)$$ for \(x \neq()\) between \(-\pi / 2\) and \(x / 2 .\) Describe the behavior of \(f(x)\) and \(g(x)\) for \(x\) close 100