Finding the limit of a multivariable function

[schniefen]

Homework Statement

Find the limit of the function below as $\textbf{x}\to\textbf{0}$.

Relevant Equations

$\frac{e^{|\textbf{x}|^2}-1}{|\textbf{x}|^2+x^2_1x_2+x^2_2x_3}$ where $\textbf{x}=(x_1,x_2,x_3)$ and $|\textbf{x}|=\sqrt{x^2_1+x^2_2+x^2_3}$.

If one approaches the origin from where $x_2=0$, the terms $x^2_1x_2+x^2_2x_3$ in the denominator equal $0$. Substituting $|\textbf{x}|^2$ for $t$ yields the expression $\frac{e^t-1}{t}$, which has limit 1 as $\textbf{x}\to\textbf{0}$ and thus $t\to0$. So the limit should be 1 if it exists. How could one show that it does exist?

 

[BvU]

How could one show that it does exist?

You need to prove that, no matter how you approach the origin, the limit will always be the same...
Try a few candidates that might yield something different, to get a feeling.

 

[schniefen]

You need to prove that, no matter how you approach the origin, the limit will always be the same...
Try a few candidates that might yield something different, to get a feeling.

Yes, but approaching along $x_1=0$ or $x_3=0$ doesn't simplify the expression, does it?

 

[BvU]

Oh, doesn't it ?
What comes out ?

 

[schniefen]

It only cancels one of the terms in the sum $x^2_1x_2+x^2_2x_3$ for $x_1=0$ and $x_3=0$ respectively.

 

[BvU]

And what is left over ?

Tip: substitute $| x| = \sqrt {...}$ to get a clearer picture, especially in the denominator .

 

[BvU]

Bedtime here (UTC+2)

 

[HallsofIvy]

With a multi-variable limit, $\left(x_1, x_2, x_3\right)$ going to (0, 0, 0), it might be best to convert to spherical coordinates. That way the limit is just as $\rho$ goes to 0. If that limit depends on $\theta$ or $\phi$ there is no limit. If it is a single number, then that is the limit.