How to show that a limit does not exist

Question

Show that the limit does not exist:

lim_{(x,y)→(1,0)} (x + y² − 1)/ √ [(x − 1)² + y⁴]

Answer 1

If we follow the given surface to the point (1,0) along two different

curves we get two different limits.

That will prove the claim.

The two different curves are

y = x-1

and

y = √(x-1)

For ease of notation, all limits below are as x->1.

That will also make y->0

When we follow the curve y = x-1

lim (x + (x-1)² - 1) / √[(x-1)² + (x-1)⁴] =

lim (x-1)x/((x-1)√[1 + (x-1)²]) =

lim x/√[1 + (x-1)²] =

1

When we follow the curve y = √(x-1)

lim (x + (x-1) - 1) / √[(x-1)² + (x-1)²] =

lim 2(x-1) / √[2(x-1)²] =

2/√2