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Chapter 2: Problem 13
Evaluate the following limits. $$\lim _{x \rightarrow \infty} \frac{\cos x^{5}}{\sqrt{x}}$$
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Suppose \(f\) is continuous at \(a\) and assume \(f(a)>0 .\) Show that there is a positive number \(\delta>0\) for which \(f(x)>0\) for all values of \(x\) in \((a-\delta, a+\delta)\). (In other words, \(f\) is positive for all values of \(x\) sufficiently close to \(a .\) )
Let $$g(x)=\left\\{\begin{array}{ll}x^{2}+x & \text { if } x<1 \\\a & \text { if } x=1 \\\3 x+5 & \text { if } x>1.\end{array}\right.$$ a. Determine the value of \(a\) for which \(g\) is continuous from the left at 1. b. Determine the value of \(a\) for which \(g\) is continuous from the right at 1. c. Is there a value of \(a\) for which \(g\) is continuous at \(1 ?\) Explain.
Find the vertical and horizontal asymptotes of \(f(x)=e^{1 / x}\).
Evaluate the following limits. $$\lim _{x \rightarrow 0^{+}} \frac{x}{\ln x}$$
a. Use the identity \(\sin (a+h)=\sin a \cos h+\cos a \sin h\) with the fact that \(\lim _{x \rightarrow 0} \sin x=0\) to prove that \(\lim _{x \rightarrow a} \sin x=\sin a\) thereby establishing that \(\sin x\) is continuous for all \(x\). (Hint: Let \(h=x-a \text { so that } x=a+h \text { and note that } h \rightarrow 0 \text { as } x \rightarrow a .)\) b. Use the identity \(\cos (a+h)=\cos a \cos h-\sin a \sin h\) with the fact that \(\lim _{x \rightarrow 0} \cos x=1\) to prove that \(\lim _{x \rightarrow a} \cos x=\cos a.\)
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