Question

Show that lim xy^3/x^2+y^2 does not exist

Answer 1

x^2+y^2=0

x^2=y^2

by equating

x=y

and let x=0

Answer 2

Correct question: Show that the limit $\lim_{(x,y) \to (0,0)}\frac{xy^3}{x^2+y^2}$ exists.

Answer:

Given,

A limit: $\lim_{(x,y) \to (0,0)}\frac{xy^3}{x^2+y^2}$.

To prove,

The given limit exists.

Solution,

The method to prove the existence of the given limit is as follows -

We know that by the Sertoz theorem when a and b are two non-negative integers and c and d are two positive even integers, the limit $\lim_{(x,y) \to (0,0)}\frac{x^ay^b}{x^c+y^d}$ will exist, if $\frac{a}{c} +\frac{b}{d} > 1$ and the value of the limit will be 0.

Here in the given limit $\lim_{(x,y) \to (0,0)}\frac{xy^3}{x^2+y^2}$, a = 1 and b = 3 are two non-negative integers and c = d = 2 are two positive even numbers.

So, $\frac{a}{c} +\frac{b}{d} = \frac{1}{2} +\frac{3}{2}=0.5+1.5=2 > 1$.

So, the limit exists and $\lim_{(x,y) \to (0,0)}\frac{xy^3}{x^2+y^2}=0$.

Hence, it is proved that the limit $\lim_{(x,y) \to (0,0)}\frac{xy^3}{x^2+y^2}$ exists.

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