Section 1
Limits
Determine the behaviour of the following functions as $x \rightarrow \pm \infty$ :(a) $f(x)=\frac{x^{3}+x+1}{2-3 x^{3}}$(b) $f(x)=\mathrm{e}^{-x} \sin x$(c) $f(x)=[x] / x$, where $[x]$ denotes the integer part of $x$
Given an $\varepsilon-\delta$ proof of the fact that$$\lim _{x \rightarrow 0} 2 x \sin \left(\frac{1}{x}\right)=0$$
Prove the quotient rule for limits of functions (see 5.1.3).
Use the rules for limits to evaluate the following.(a) $\lim _{x \rightarrow 0} \frac{x^{3}-1}{x^{2}+1}$(b) $\lim _{x \rightarrow 1} \frac{x-1}{x^{3}}-\frac{1}{-1}$(c) $\lim _{x \rightarrow 1} \cos \left(\frac{\pi x}{x+1}\right)$(d) $\lim _{x \rightarrow 0} x^{2} \cos \left(\frac{1}{x}\right)$
Evaluate the following one-sided limits.(a) $\lim _{x \rightarrow 1-} \frac{1}{[x]+1}$(b) $\lim _{x \rightarrow 4+}[\sqrt{x}]$(c) $\lim _{x \rightarrow 4-}[\sqrt{x}]$
Let $f(x)$ be defined on an interval $(0, a)$ and let $t=1 / x$. Prove that if either of the limits $\lim _{x \rightarrow 0+} \int(x)$ or $\lim _{t \rightarrow \infty} f(1 / t)$ exists then both exist and have the same value.