202009272336 Homework 1 (Q3)

Verify the following statements by definition of limit.

[Hint: Use the -language.]

i. $\displaystyle{\lim_{n\to \infty}\frac{2n^2-1}{4n^2+2}=\frac{1}{2}}$ ;

ii. $\displaystyle{\lim_{n\to \infty}\frac{\sqrt{n^2+n}}{n}=1}$ ;

iii. $\displaystyle{\lim_{n\to \infty}\sqrt[n]{n+2}=1}$ .

Verification.

i. Given

$\displaystyle{\lim_{n\to \infty}\frac{2n^2-1}{4n^2+2}=\frac{1}{2}}$ ,

or,

$\displaystyle{\lim_{n\to \infty}f(n)=\frac{1}{2}}$ ,

in which

$\displaystyle{f(n)=\frac{2n^2-1}{4n^2+2}}$ .

Informally speaking of it, the function $f(n)$ shall go to the limit $1/2$ if its variable $n$ goes to the ( $+\textrm{ve}$ ) infinity.

Remark.

In other words, if a function $f$ tends to its limit $L$ , by that it is to say, there exists some output value(s) $f(x)$ sufficiently close to the number $L$ .

Make-up arguments.

As needs

,

so let

Roughwork. (Instructive)

,

s.t.

(to be continued)

Definition. (Limit) Let $f$ be a real-valued function defined on a subset $D$ of the real numbers $\mathbb{R}$ . Let $c$ be a limit point of $D$ and let $L$ be a real number. Symbolically:

$\begin{aligned} & \qquad\enspace \lim_{x\to c} f(x) = L \\ & \Longleftrightarrow \bigg( \forall\, \epsilon >0,\, \exists\, \delta >0,\, \forall\, x\in D, 0<|x-c|<\delta \Rightarrow \big| f(x)-L \big| < \epsilon \bigg) \\ \end{aligned}$