evaluate a multivariate limit

Try each step and move on to the next if indeterminate form is encountered:

Step 1 - Simple Substitution

Try just substituting the coordinate of the limit $(x,y,\dots )$ into the expression.

Step 2 - Algebraic Manipulation

Factoring

If the expression is a fraction, try factoring the numerator and denominator and see if anything cancels. Then evaluate the limit again.

Multiply by conjugate over conjugate

If the fraction contains a root in numerator/denominator (e.g. in the form $a+\sqrt{b}$), try getting rid of it by multiplying the fraction by the conjugate over conjugate. The conjugate for $a+\sqrt{b}$ is $a-\sqrt{b}$. This might get rid of indeterminate forms if you are lucky.

Try to find other ways

See if there’s any other things you can try, like trignometry identities, to make the expression evaluatable.

Step 3 - Evaluate along paths to prove limit DNE

It’s likely that the limit does not exist, but we need to prove it. Try approaching the limit along different paths, like $x=0$ (y-axis), $y=0$ (x-axis), $y=x$, $y=x^3$, etc. Substitute the value into the limit expression then evaluate the limit. If any two limits evaluated along different paths are not equal to each other, then the limit does not exist. Note that if two limits evaluated along different paths end up being equal, this number is not necessarily the limit of the function.