Show that the limit exists or does not exist

Question

$\lim_{(x, y) \to (0,0)} \frac{5x^2}{x^2 + y^2}$

let $y = 0$

$\lim_{x \to 0} \frac{5x^2}{x^2} = 5$

let $y = x$

$\lim_{x \to 0} \frac{5x^2}{2x^2} = \frac{5}{2}$

Since different values the limit does not exist.

Would this be right?

Answer 1

That’s right: if the limit exists, all the restriction of the function have the same limit. This is not the case, hence the limit does not exist.

Answer 2

Yes that's right, more in general we can see that by $y=mx$

$$\frac{5x^2}{x^2 + y^2}=\frac{5}{1+m^2}$$

or by polar coordinates

$$\frac{5x^2}{x^2 + y^2}=5\cos^2\theta$$

which depend respectively upon $m$ and $\theta$.

You can find many others similar questions on MSE, as for example

• Calculate the limit given.
• Limit with two-variable function