Valid method of evaluating limit?

[Defennder]

Homework Statement

I don't know if the following is valid, so I'll appreciate if someone could tell me if it's ok. I want to find
$$\lim_{x\rightarrow \infty} \frac{x}{\sqrt{a^2+x^2}}$$

Homework Equations

The Attempt at a Solution

Using L'Hopital rule doesn't appear to help, because repeatedly differentiating both the top and bottom gives the same limit. So I did a substitution:

$$x=a \tan \theta$$
And the problem now becomes $$\lim_{\theta \rightarrow \frac{\pi}{2}} \sin \theta$$ which easily evaluates to 1. Is this correct?

 

[Zizy]

Physical method would be: a^2 is small compared to infinity, so we can ignore it. Sqrt(x^2)=x, so limit is 1 :P
But yes, I guess your method is just fine.

 

[Dick]

It's valid, but why would you want to do it that way? Just divide numerator and denominator by x and use algebra. Which is what Zizy is saying.

 

[Defennder]

OK thanks. I got confused for a moment over something.