91Ó°ÊÓ

$$\lim _{x \rightarrow 0} \frac{1-\cos x}{x \sin x}$$

Use the formal definitions of limits as \(x \rightarrow \pm \infty\) to establish the limits If \(f\) has the constant value \(f(x)=k,\) then \(\lim _{x \rightarrow-\infty} f(x)=k\)

$$\lim _{x \rightarrow 0} \frac{2 x^{2}}{3-3 \cos x}$$

Use formal definitions to prove the limit statements. $$\lim _{x \rightarrow 0} \frac{-1}{x^{2}}=-\infty$$

Use formal definitions to prove the limit statements. $$\lim _{x \rightarrow 3} \frac{-2}{(x-3)^{2}}=-\infty$$

Use formal definitions to prove the limit statements. $$\lim _{x \rightarrow-5} \frac{1}{(x+5)^{2}}=\infty$$

Here is the definition of infinite right-hand limit. We say that \(f(x)\) approaches infinity as \(x\) approaches \(c\) from the right, and write $$\lim _{x \rightarrow c^{+}} f(x)=\infty$$ if, for every positive real number \(B\), there exists a corresponding number \(\delta>0\) such that for all \(x\) $$cB$$ Modify the definition to cover the following cases. a. \(\lim _{x \rightarrow c^{-}} f(x)=\infty\) b. \(\lim _{x \rightarrow c^{+}} f(x)=-\infty\) c. \(\lim _{x \rightarrow c^{-}} f(x)=-\infty\)

Graph the curves. Explain the relationship between the curve's formula and what you see. $$y=\frac{x}{\sqrt{4-x^{2}}}$$

Graph the curves. Explain the relationship between the curve's formula and what you see. $$y=\frac{-1}{\sqrt{4-x^{2}}}$$

Graph the curves. Explain the relationship between the curve's formula and what you see. $$y=\sin \left(\frac{\pi}{x^{2}+1}\right)$$