Answer to Question #135765 in Calculus for Promise Omiponle

Question #135765
Prove that the following limit does not exist:

lim(x,y)->(1,1) (xy^(2)-1)/(y-1)
1
Expert's answer
2020-10-01T13:57:15-0400

lim(x,y)(1,1)xy21y1\lim\limits_{(x,y)\rightarrow(1,1)}\frac{xy^2-1}{y-1}

To show that the limit does not exist take two different pairs of sequences with limit 1

1, and show that the limit in both cases is not equal.

if we set x = y we get

limy1y31y1=limy1(y2+y+1)=3\lim_{y\to 1} \frac{y^3-1}{y-1}=\lim_{y\to 1}(y^2+y+1)=3

if we set x = 1 we get

limy1y21y1=limy1(y+1)=2\lim_{y\to 1} \frac{y^2-1}{y-1}=\lim_{y\to 1}(y+1)=2

So the limit does not exist.

Answer: the statement that the limit does not exist is true



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