[Math] How to prove infinite limit is limit does not exist using epsilon and delta

Question

So I was recently taught that If $\lim_{x→0}f(x)=∞$, then the limit does not exist, can anyone explain that using epsilon and delta if its possible? But honestly any sort of explanation would be fine

Answer 1

$\lim_{x→0}f(x)=∞$is defined as

For every $M>0$ there exits a $\delta >0$ such that if $0<|x|<\delta$ then $f(x)>M$

That simply means we can make $f(x)$ as large as we wish but the price to pay is to make |x| small enough.

For example we can make $\frac {1}{x^2}$ larger than $10000$ provided that we make $|x|$ less than $0.01$