Why does the limit of 1/x² not exist as x approaches zero?

[Hwng10]

Homework Statement

Show limit 1/x^2 when x approaches zero does not exist.

Homework Equations

The Attempt at a Solution

What I do is suppose the limit exists,say L.Then I show that for all real number L, the limit does not approaches L. In this case, I separate L into different cases. Can anyone help me to check is this a valid prove.

 

[HallsofIvy]

Yes, that is a valid approach.

 

[Mark44]

$$\lim_{x \to 0}\frac{1}{x^2} = \infty$$

Since ∞ is not a real number the limit above is said not to exist. This limit means that for any large, positive number M, there is some interval around 0 such that if x is in that interval, 1/x² > M.