Question

What is the limit as x approaches 0 of `1/x^2`?

Answer 1

Infinity `+oo`

Explanation:

It is easily shown, that, as `x` gets smaller, `x^2` gets smaller at an even greater rate, so `1/x^2` will be greater.

A few steps:

`x=1->x^2=1->1/x^2=1`

`x=1/2->x^2=1/4->1/x^2=4`

`x=1/100->x^2=10000->1/x^2=10000`

This means that the closer `x` goes to `0` the higher the function goes. In this case it doesn't matter whether `x->0` from the positive side or from the negative, as the square makes it al positive. By choosing smaller and smaller values of `x`, the function can reach any size you want.

Translated to "the language":

`lim_(x->0^+) 1/x^2=lim_(x->0^-) 1/x^2=lim_(x->0) 1/x^2= oo`
graph{1/x^2 [-17.75, 18.3, -1.61, 16.42]}