91Ó°ÊÓ

Each of Exercises \(15-30\) gives a function \(f(x)\) and numbers \(L, c,\) and \(\varepsilon>0 .\) In each case, find the largest open interval about \(c\) on which the inequality \(|f(x)-L|<\varepsilon\) holds. Then give a value for \(\delta>0\) such that for all \(x\) satisfying \(0<|x-c|<\delta,\) the inequality \(|f(x)-L|<\varepsilon\) holds. $$f(x)=\sqrt{x}, \quad L=1 / 2, \quad c=1 / 4, \quad \varepsilon=0.1$$

Find the limits. a. \(\lim _{x \rightarrow 0^{+}} \frac{|\sin x|}{x}\) b. \(\lim _{x \rightarrow 0^{-}} \frac{|\sin x|}{x}\)

Each of Exercises \(15-30\) gives a function \(f(x)\) and numbers \(L, c,\) and \(\varepsilon>0 .\) In each case, find the largest open interval about \(c\) on which the inequality \(|f(x)-L|<\varepsilon\) holds. Then give a value for \(\delta>0\) such that for all \(x\) satisfying \(0<|x-c|<\delta,\) the inequality \(|f(x)-L|<\varepsilon\) holds. $$f(x)=\sqrt{19-x}, \quad L=3, \quad c=10, \quad \varepsilon=1$$

Find the limits. $$\lim _{y \rightarrow-3}(5-y)^{4 / 3}$$

Find the limit of each rational function (a) as \(x \rightarrow \infty\) and (b) as \(x \rightarrow-\infty .\) Write \(\infty\) or \(-\infty\) where appropriate. $$g(x)=\frac{10 x^{5}+x^{4}+31}{x^{6}}$$

Find the limit of each rational function (a) as \(x \rightarrow \infty\) and (b) as \(x \rightarrow-\infty .\) Write \(\infty\) or \(-\infty\) where appropriate. $$g(x)=\frac{x^{3}+7 x^{2}-2}{x^{2}-x+1}$$

Find the limits. $$\lim _{z \rightarrow 4} \sqrt{z^{2}-10}$$

Gives a function \(f(x)\) and numbers \(L, c,\) and \(\varepsilon>0 .\) In each case, find the largest open interval about \(c\) on which the inequality \(|f(x)-L|<\varepsilon\) holds. Then give a value for \(\delta>0\) such that for all \(x\) satisfying \(0<|x-c|<\delta,\) the inequality \(|f(x)-L|<\varepsilon\) holds. $$f(x)=\sqrt{x-7}, \quad L=4, \quad c=23, \quad \varepsilon=1$$

Find the limits. a. \(\lim _{x \rightarrow 0^{+}} \frac{1-\cos x}{|\cos x-1|}\) b. \(\lim _{x \rightarrow 0} \frac{\cos x-1}{|\cos x-1|}\)

At what points are the functions continuous? $$y=\frac{x+2}{\cos x}$$