MHB Is Absolute Value Necessary for Proving Limit with Epsilon for $\frac{1}{x^4}$?

[Dethrone]

Prove that $\lim_{{x}\to{0}}\frac{1}{x^4}=\infty$, given a $M>0$

So we need to prove that $f(x) > M$:

$\frac{1}{x^4}>M$, $\frac{1}{M}>x^4$, $\frac{1}{M^{1/4}}>|x|$

Is that right so far? Is the absolute values necessary in my last statement?

 

[mathbalarka]

That looks good to me. All you have to do now is to pick up a positive $\delta < 1/M^{1/4}$ such that for all positive real $x < \delta$, $1/x^4 > M$.

Is the absolute values necessary in my last statement?

Well, at least you have to state explicitly that $x$ is a positive real before drawing that argument.

 

[Ackbach]

I would say yes, because it is a two-sided limit.