Question

What is the limit of `lim_((x,y)->(0,0))x^2/(x^2+y^2)`?

Answer 1

The limit does not exist.

Explanation:

In order for this limit to exist, the fraction `x^2/(x^2+y^2)` must approach the same value `L`, regardless of the path along which we approach `(0,0).`

Consider approaching `(0,0)` along the `x`-axis. That means fixing `y=0` and finding the limit `lim_(x->0) x^2/(x^2+y^2).` We get

`lim_{x->0," "y=0} x^2/(x^2+y^2)=lim_(x->0)x^2/(x^2+0)`

`color(white)(lim_{x->0," "y=0} x^2/(x^2+y^2))=lim_(x->0)x^2/x^2`
`color(white)(lim_{x->0," "y=0} x^2/(x^2+y^2))=lim_(x->0)1`
`color(white)(lim_{x->0," "y=0} x^2/(x^2+y^2))=1`

Now, consider approaching `(0,0)` along the `y`-axis. This means fixing `x=0` and finding the limit `lim_(y->0) x^2/(x^2+y^2).` We get

`lim_{y->0," "x=0} x^2/(x^2+y^2)=lim_(y->0)0/(0+y^2)`
`color(white)(lim_{y->0," "x=0} x^2/(x^2+y^2))=lim_(y->0)0/(y^2)`
`color(white)(lim_{y->0," "x=0} x^2/(x^2+y^2))=lim_(y->0)0`
`color(white)(lim_{y->0," "x=0} x^2/(x^2+y^2))=0`

Approaching the origin along these two different paths leads to different limits.

`lim_{x->0," "y=0} x^2/(x^2+y^2)!=lim_{y->0," "x=0} x^2/(x^2+y^2)`

Thus, the original limit does not exist.