91Ó°ÊÓ

Let \(f:(0, \infty) \rightarrow \mathbb{R}\). Show that \(\lim _{x \rightarrow \infty} f(x)=L\) if and only if \(\lim _{x \rightarrow 0+} f(1 / x)=L\)

What are the limits \(\lim _{x \rightarrow \infty} x^{p}\) for various real numbers \(p\) ?

Show that the characteristic function of the rationals can also be defined by the formula $$ \chi_{\mathbb{Q}}(x)=\lim _{m \rightarrow \infty} \lim _{n \rightarrow \infty}|\cos (m ! \pi x)|^{n} $$

Show that one of the limits \(\lim _{x \rightarrow 0+} f(x)\) and \(\lim _{x \rightarrow 0-} f(x)\) of the function $$ f(x)=e^{1 / x} $$ at 0 is infinite and one is finite. What can you say about the limits $$ \lim _{x \rightarrow \infty} f(x) \text { and } \lim _{x \rightarrow-\infty} f(x) ? $$

Show that $$ \lim _{x \rightarrow x_{0}+} \chi_{\mathbb{Q}}(x) \text { and } \lim _{x \rightarrow x_{0}-} \chi_{\mathbb{Q}}(x) $$ both fail to exist, where \(\chi_{\mathbb{Q}}\) is the characteristic function of the rationals. What would be the answer to the corresponding question for the characteristic function of the irrationals?

Give an example of a set \(E\) such that the characteristic function \(\chi_{E}\) of \(E\) has limits at every point. Can you describe the most general set \(E\) with this property?

Give an example of a set \(E\) such that the characteristic function \(\chi_{E}\) of \(E\) has one-sided limits at every point. Can you describe the most general set \(E\) with this property?

Sketch the graph of the characteristic function \(\chi_{K}\) of the Cantor set (Exercises \(4.3 .23\) and \(4.4 .9)\) and show that $$ \lim _{x \rightarrow x_{0}} \chi_{K}(x)=0 $$ at all points \(x\) not in the Cantor set and that this limit fails to exist at all points in Cantor set.