# Recitation 3

Problem 1: Show $\lim_{x\to\infty} [x]/x = 1$.

Idea: Use $x-1<[x]\le x$ and the sandwich theorem.

Problem 2: Given sets $A$ and $B$. It is known that for every $a\in A$ there exists $b\in B$ such that $a < b - 1/10000$. Show that $\sup A < \sup B$.

Proof: By the definition of supreme, there is $a\in A$ such that $a > \sup A - 1/10000$. By the condition in the problem, there is $b\in B$ such that $a < b - 1/10000$. Thus $\sup A < b \le \sup B$.

Remark: If for every $a\in A$ there exists $b\in B$ such that $a < b$, we can only conclude that $\sup A \le \sup B$ (why?). In general, we cannot conclude $\sup A < \sup B$. For example, $A = (0, 1)$ and $B = \{1\}$.

Problem 3: Find two functions $f$ and $g$ such that $\lim_{x\to \infty}f(x) = \lim_{x\to \infty}g(x) = \infty$, but $\lim_{x\to \infty}f(x)/g(x)$ does not exist (even in the broad sense).

Idea: Take $h(x) = \sin x + 2$ (or any other positive function which does not have a limit as $x$ approaches infinity). Notice that $h(x) \ge 1$ for all $x$. Take $g(x) = x$ (or any other function whose limit at infinity is $\infty$). Now let $f(x) = g(x)h(x)$.

Problem 4: Suppose $\lim_{x\to \infty}f(x) = \infty$ and $\lim_{x\to \infty}f(x)g(x) = 1$. Show that necessarily $\lim_{x\to \infty}g(x) = 0$.

Sketch: Our goal is to prove that for every $\epsilon > 0$ there is $M$ such that $|g(x)|<\epsilon$ for all $x>M$. We can find $M_1$ such that $f(x) > 2/\epsilon$ for all $x>M_1$ and $M_2$ such that $0 < f(x)g(x) < 2$. Now take $M = \max(M_1, M_2)$.

Problem 5: Compute the following limits (a) $\lim_{x\to \infty}\sqrt{x+1}-\sqrt{x}$; (b) $\lim_{x\to \infty}\frac{x+\sin x}{x+\cos x}$; (c) $\lim_{x\to a}\frac{\sin^2 x - \sin^2 a}{x^2 - a^2}$; (d) $\lim_{x\to \infty}\sin 2x / \sin 3x$; (e) $\lim_{x\to 1}\frac{x^n-1}{x^m-1}$; (f) $\lim_{x\to \infty}\frac{x^4-5}{x^2+4x+7}$; (g)$\lim_{x\to \infty}\frac{x^3}{x^2+1}-x$.

Answers: (a) 0; (b) 1; (c) $\frac{\sin a}{a}$ if $a\neq 0$; 1 if $a = 0$; (d) 2/3; (e) $n/m$; (f) $\infty$; (g) 0.