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Chapter 8: Problem 1
Calculate \(\cos (.2), \sin (.2)\) and \(\cos 1, \sin 1\) correct to four decimal places.
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(a) Show that if \(\alpha>0\), then the function \(x \mapsto x^{\alpha}\) is strictly increasing on \((0, \infty)\) to \(\mathbb{R}\) and that \(\lim _{x \rightarrow 0+} x^{\alpha}=0\) and \(\lim _{x \rightarrow \infty} x^{\alpha}=\infty\) (b) Show that if \(\alpha<0\), then the function \(x \mapsto x^{\alpha}\) is strictly decreasing on \((0, \infty)\) to \(\mathbb{R}\) and that \(\lim _{x \rightarrow 0+} x^{\alpha}=\infty\) and \(\lim _{x \rightarrow \infty} x^{\alpha}=0\).
Show that if \(x>0\) then $$ 1-\frac{x^{2}}{2}+\frac{x^{4}}{24}-\frac{x^{6}}{720} \leq \cos x \leq 1-\frac{x^{2}}{2}+\frac{x^{4}}{24} $$ Use this inequality to establish a lower bound for \(\pi\).
Show that \(\lim \left((\cos \pi x)^{2 n}\right)\) exists for all \(x \in \mathbb{R}\). What is its limit?
Let \(f_{n}(x):=n x /\left(1+n x^{2}\right)\) for \(x \in A:=[0, \infty)\), Show that each \(f_{n}\) is bounded on \(A_{1}\) but the pointwise limit \(f\) of the sequence is not bounded on \(A\). Does \(\left(f_{n}\right)\) converge uniformly to \(f\) on \(A\) ?
Evaluate \(L^{\prime}(1)\) by using the sequence \((1+1 / n)\) and the fact that \(e=\lim \left((1+1 / n)^{n}\right)\).
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