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Chapter 6: Problem 2
Show that \(f(x):=x^{1 / 3}, x \in \mathbb{R}\), is not differentiable at \(x=0\).
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If \(g(x):=\sin x\), show that the remainder term in Taylor's Theorem converges to zero as \(n \rightarrow \infty\) for each fixed \(x_{0}\) and \(x\).
Evaluate the following limits: (a) \(\lim _{x \rightarrow 0} \frac{\operatorname{Arctan} x}{x}(-\infty, \infty)\), (b) \(\lim _{x \rightarrow 0} \frac{1}{x(\ln x)^{2}} \quad(0,1)\), (c) \(\lim _{x \rightarrow 0+} x^{3} \ln x \quad(0, \infty)\) (d) \(\lim _{x \rightarrow \infty} \frac{x^{3}}{e^{x}}(0, \infty)\).
Evaluate the following limits: (a) \(\lim _{x \rightarrow \infty} x^{1 / x} \quad(0, \infty)\). (b) \(\lim _{x \rightarrow 0+}(\sin x)^{x}(0, \pi)\). (c) \(\lim _{x \rightarrow 0+} x^{\sin x}(0, \infty)\), (d) \(\lim _{x \rightarrow \pi / 2-}(\sec x-\tan x) \quad(0 . \pi / 2)\)
Let \(f(x):=\cos a x\) for \(x \in \mathbb{R}\) where \(a \neq 0\). Find \(f^{(n)}(x)\) for \(n \in \mathbb{N}, x \in \mathbb{R}\).
Given that the restriction of the tangent function tan to \(I:=(-\pi / 2, \pi / 2)\) is strictly increasing and that \(\tan (I)=\mathbb{R}\), let Arctan: \(\mathbb{R} \rightarrow \mathbb{R}\) be the function inverse to the restriction of \(\tan\) to \(I\). Show that Arctan is differentiable on \(\mathbb{R}\) and that \(D \operatorname{Arctan}(y)=\left(1+y^{2}\right)^{-1}\) for \(y \in \mathbb{R}\).
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