Proving a limit does not exist

The two path method works for limits of multiple variables. One first makes various substitutions then finds the limit of each new functions in hopes to find two with different limits, thus proving that the limit does not exist.

using substitution :

using substitution

Thus by the two limit method the above limit does not exist. Note: this method can only be used to prove a limit does NOT exist, you need to use a different method to prove that a limit does exist.

See:

• Multivariable Calculus Notes - Chapter 14 - Sections 1-2