91Ó°ÊÓ

Gives a function \(f(x)\) and numbers \(L, c,\) and \(\epsilon \geq 0 .\) In each case, find an open interval about \(c\) on which the inequality \(|f(x)-L|<\epsilon\) holds. Then give a value for \(\delta>0\) such that for all \(x\) satisfying \(0<|x-c|<\delta\) the inequality \(|f(x)-L|<\epsilon\) holds. \(f(x)=2 x-2, \quad L=-6, \quad c=-2, \quad \epsilon=0.02\)

Find the limits. $$\lim _{s \rightarrow 2 / 3}(8-3 s)(2 s-1)$$

Find the limits. $$\lim _{h \rightarrow 0^{-}} \frac{\sqrt{6}-\sqrt{5 h^{2}+11 h+6}}{h}$$

Find the limit of each rational function (a) as \(x \rightarrow \infty\) and \((b)\) as \(x \rightarrow-\infty\). $$f(x)=\frac{3 x+7}{x^{2}-2}$$

Find the limit of each rational function (a) as \(x \rightarrow \infty\) and \((b)\) as \(x \rightarrow-\infty\). $$h(x)=\frac{7 x^{3}}{x^{3}-3 x^{2}+6 x}$$

Find the limits. $$\lim _{x \rightarrow-1 / 2} 4 x(3 x+4)^{2}$$

Find the limits. $$\lim _{x \rightarrow-2^{+}}(x+3) \frac{|x+2|}{x+2}$$

Gives a function \(f(x)\) and numbers \(L, c,\) and \(\epsilon \geq 0 .\) In each case, find an open interval about \(c\) on which the inequality \(|f(x)-L|<\epsilon\) holds. Then give a value for \(\delta>0\) such that for all \(x\) satisfying \(0<|x-c|<\delta\) the inequality \(|f(x)-L|<\epsilon\) holds. \(f(x)=\sqrt{x+1}, \quad L=1, \quad c=0, \quad \epsilon=0.1\)

Find the limit of each rational function (a) as \(x \rightarrow \infty\) and \((b)\) as \(x \rightarrow-\infty\). $$h(x)=\frac{9 x^{4}+x}{2 x^{4}+5 x^{2}-x+6}$$

Gives a function \(f(x)\) and numbers \(L, c,\) and \(\epsilon \geq 0 .\) In each case, find an open interval about \(c\) on which the inequality \(|f(x)-L|<\epsilon\) holds. Then give a value for \(\delta>0\) such that for all \(x\) satisfying \(0<|x-c|<\delta\) the inequality \(|f(x)-L|<\epsilon\) holds. \(f(x)=\sqrt{x}, \quad L=1 / 2, \quad c=1 / 4, \quad \epsilon=0.1\)