Problem 1
In the following cases show that \(V_{f}[I]=+\infty,\) though \(f\) is bounded on
I. (In case (iii), \(f\) is continuous, and in case (iv), it is even
differentiable on \(I .)\)
$$
\text { (i) For } I=[a, b](a
Problem 2
Show that \(f\) is absolutely continuous (in the weaker sense) on \([a, b]\) if for every \(\varepsilon>0\) there is \(\delta>0\) such that $$ \begin{array}{c} \sum_{i=1}^{m}\left|f\left(t_{i}\right)-f\left(s_{i}\right)\right|<\varepsilon \text { whenever } \sum_{i=1}^{m}\left(t_{i}-s_{i}\right)<\delta \text { and } \\\ a \leq s_{1} \leq t_{1} \leq s_{2} \leq t_{2} \leq \cdots \leq s_{m} \leq t_{m} \leq b \end{array} $$ (This is absolute continuity in the stronger sense.)
Problem 4
Why does \(\lim _{x \rightarrow+\infty} \frac{f(x)}{g(x)}\) not exist, though \(\lim _{x \rightarrow+\infty} \frac{f^{\prime}(x)}{g^{\prime}(x)}\) does, in the fol- lowing example? Verify and explain. $$ f(x)=e^{-2 x}(\cos x+2 \sin x), \quad g(x)=e^{-x}(\cos x+\sin x) $$ [Hint: \(g^{\prime}\) vanishes many times in each \(G_{+\infty}\). Use the Darboux property for the proof.]
Problem 5
Find \(\lim _{x \rightarrow 0^{+}} \frac{e^{-1 / x}}{x}\). [Hint: Substitute \(z=\frac{1}{x} \rightarrow+\infty\). Then use the rule.]
Problem 6
Prove that if the functions \(f_{i}: E^{1} \rightarrow E^{*}(C)\) are differentiable at \(p,\) so is their product, and $$ \left(f_{1} f_{2} \cdots f_{m}\right)^{\prime}=\sum_{i=1}^{m}\left(f_{1} f_{2} \cdots f_{i-1} f_{i}^{\prime} f_{i+1} \cdots f_{m}\right) \text { at } p $$
Problem 6
Verify that the assumptions of L'Hôpital's rule hold, and find the following limits. (a) \(\lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}}{\ln (e-x)+x-1} ;\) (b) \(\lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}-2 x}{x-\sin x}\) (c) \(\lim _{x \rightarrow 0} \frac{(1+x)^{1 / x}-e}{x} ;\) (d) \(\lim _{x \rightarrow 0^{+}}\left(x^{q} \ln x\right), q>0\) (e) \(\lim _{x \rightarrow+\infty}\left(x^{-q} \ln x\right), q>0\) (f) \(\lim _{x \rightarrow 0^{+}} x^{x}\) (g) \(\lim _{x \rightarrow+\infty}\left(x^{q} a^{-x}\right), a>1, q>0\) (h) \(\lim _{x \rightarrow 0}\left(\frac{1}{x^{2}}-\operatorname{cotan}^{2} x\right)\); (i) \(\lim _{x \rightarrow+\infty}\left(\frac{\pi}{2}-\arctan x\right)^{1 / \ln x}\) (j) \(\lim _{x \rightarrow 0}\left(\frac{\sin x}{x}\right)^{1 /(1-\cos x)}\).
Problem 6
Let \(g: E^{1} \rightarrow E^{1}\) (real) and \(f: E^{1} \rightarrow E\) be relatively continuous on \(J=[c, d]\) and \(I=[a, b],\) respectively, with \(a=g(c)\) and \(b=g(d) .\) Let $$ h=f \circ g $$ Prove that if \(g\) is one to one on \(J,\) then (i) \(g[J]=I,\) so \(f\) and \(h\) describe one and the same arc \(A=f[I]=h[J]\); (ii) \(V_{f}[I]=V_{h}[J] ;\) i.e., \(\ell_{f} A=\ell_{h} A .\) [Hint for (ii): Given \(P=\left\\{a=t_{0}, \ldots, t_{m}=b\right\\}\), show that the points \(s_{i}=g^{-1}\left(t_{i}\right)\) form a partition \(P^{\prime}\) of \(J=[c, d],\) with \(S\left(h, P^{\prime}\right)=S(f, P) .\) Hence deduce \(V_{f}[I] \leq\) \(V_{h}[J]\) Then prove that \(V_{h}[J] \leq V_{f}[I],\) taking an arbitrary \(P^{\prime}=\left\\{c=s_{0}, \ldots, s_{m}=d\right\\},\) and defining \(P=\left\\{t_{0}, \ldots, t_{m}\right\\},\) with \(t_{i}=g\left(s_{i}\right) .\) What if \(\left.g(c)=b, g(d)=a ?\right]\)
Problem 6
Show that \(f^{\prime}\) need not be continuous or bounded on \([a, b]\) (under the standard metric), even if \(f\) is differentiable there.
Problem 7
For any \(s \in E^{1}\) and \(n \in \bar{N},\) define $$ \left(\begin{array}{l} s \\ n \end{array}\right)=\frac{s(s-1) \cdots(s-n+1)}{n !} \text { with }\left(\begin{array}{l} s \\ 0 \end{array}\right)=1 \text { . } $$ Then prove the following. (i) \(\lim _{n \rightarrow \infty} n\left(\begin{array}{l}s \\\ n\end{array}\right)=0\) if \(s>0\). (ii) \(\lim _{n \rightarrow \infty}\left(\begin{array}{l}s \\\ n\end{array}\right)=0\) if \(s>-1\). (iii) For any fixed \(s \in E^{1}\) and \(x \in(-1,1),\) $$ \lim _{n \rightarrow \infty}\left(\begin{array}{l} s \\ n \end{array}\right) n x^{n}=0 $$ hence $$ \lim _{n \rightarrow \infty}\left(\begin{array}{l} s \\ n \end{array}\right) x^{n}=0 $$
Problem 7
Prove that $$ \int_{0}^{x} \frac{\ln (1-t)}{t} d t=\sum_{n=1}^{\infty} \frac{x^{n}}{n^{2}} \quad \text { for } x \in[-1,1] $$
