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Given the following limit:

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

And the instrucion to "Determine whether the limit exists, give a complete argument", would the following be a "complete argument"?

Approaching the limit from the line y=0, gives $ \lim_{(x,y)\to (0,0)} \frac{xy^2}{x^3+y^3} = \lim_{(x,y)\to (0,0)} \frac{0}{x^3+y^3} = 0 $

Approaching the limit from the line y=x, gives $ \lim_{(x,y)\to (0,0)} \frac{xy^2}{x^3+y^3} = \lim_{(x,y)\to (0,0)} \frac{x^3}{2x^3} = \frac{1}{2} $

These limits do not agree, thus the original limit $$ \lim_{(x,y)\to (0,0)} \frac{xy^2}{x^3+y^3} $$ does not exist.

Or should another method besides approaching from different lines be used to give a "complete argument" whether this limit exists be given?

Community
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B_s
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2 Answers2

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This is exactly the way you should approach the problem. Showing that the limit takes on different values depending on what path you use to approach is a sufficient condition for a multi-variable limit to not exist.

apnorton
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Maybe this is helpful:

In fact, the limit $\lim_{(x,y)\to (a,b)}f(x,y)$ exists and equals some value like $c$ if and only for any sequence $\{x_n\}_{n}$ converging to $a$ and any sequence $\{y_n\}_{n}$ converging to $b$ one has that $$\lim_{n\to \infty}f(x_n,y_n)=c.$$

So for proving of non-existence of $\lim_{(x,y)\to (a,b)}f(x,y)$, it is enough to list two sequences $\{(x_n,y_n)\}_{n}$ and $\{(x'_n,y'_n)\}_{n}$ both converging to $(a,b)$ while $$\lim_{n\to \infty}f(x_n,y_n)\ne\lim_{n\to \infty}f(x'_n,y'_n).$$

Note that here you have almost done the same by mentioning $x_n=a_{n}$ and $y_n=0$, and also $x'_n=a_{n}$ and $y'_n=a_{n}$, where $\{a_n\}_n$ is an arbitrary sequence such that $a_{n}\to 0$, e.g. like $a_n=\frac{1}{n} $

Mohammad Khosravi
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