Epsilon-Delta?

Prove using "Epsilon-Delta" definition of two-sided limit:

$\large{ \lim_{x \rightarrow 3}{x^2}=9}$

Definition Let $f(x)$ be defined for all $x$ in some open containing the number $a$ with possible exception that $f(x)$ need not to be defined at $a$ . We will write $\large{ \lim_{x \rightarrow a}{f(x)}=L}$ if given any number $\epsilon > 0$ we can find a number $\delta > 0$ such that $\large{ |f(x)-L|< \epsilon \quad \text{if} \quad 0 < |x-a|< \delta}$

#Calculus

Easy Math Editor

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Comments

Like most problems, I'm doing those in a somewhat "un-orthodox" way.

Given $\epsilon>0$ . If $\epsilon\leq 1$ , let $\delta=\frac{\epsilon}{12}$ . Now $3-\frac{\epsilon}{12}<x<3+\frac{\epsilon}{12}$ implies $9-\frac{\epsilon}{2}+\frac{\epsilon^2}{12^2}<x^2<9+\frac{\epsilon}{2}+\frac{\epsilon^2}{12^2}$ so that $|x^2-9|<\epsilon$ as required. If $\epsilon>1$ let $\delta=\frac{1}{12}$ .

Otto Bretscher - 5 years, 3 months ago

@Otto Bretscher, @Pi Han Goh, @Calvin Lin

Department 8 - 5 years, 3 months ago

Epsilon-delta definition of a limit

Pi Han Goh - 5 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$