Evaluate the limit of a function

Question

^{Does lim}(x⁴– 4y² )/(x²+2y²) as (x,y)→(0,0) exist? Explain your answer.

Answer 1

The easiest way to show that the answer is no is to let each of x and y = 0 and evaluate.

When x = 0, we get -4y^2/2y^2 = -2 for all y not equal to 0, including as y goes to 0.

When y = 0, we get x^4/x^2 = x^2 for x not equal to 0, which has a limit of 0 as x goes to 0.

The two different limits, -2 and 0, tell us that there is no limit.