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.