Limits for x approaching infinity

[funnytim]

Hello everyone!

I'm having problems with a calculus problem here,

1) Solve lim (x-->[infinity]) [(square root x^2 + x + 1)] / x

I never did quite understand how to do it if its limit as x approaches infinity. I believe I have to find a way to get ride of the bottom x ?

Thanks!

 

[galactus]

$\displaystyle \lim_{x\to \infty}\frac{\sqrt{x^{2}+x+1}}{x}$

The trick is dividing the top by $\displaystyle \sqrt{x^{2}}$ and the bottom by x.

Yes, we can do that.

Divide the top by $\displaystyle \sqrt{x^{2}}$ and the bottom by x gives us $\displaystyle \lim_{x\to \infty}\frac{\sqrt{1+\frac{1}{x}+\frac{1}{x^{2}}}}{1}$

Now, see the limit?.

 

[BigGlenntheHeavy]

$\displaystyle \lim_{x\to\infty}\frac{\sqrt(x^{2}+x+1)}{x} \ = \ \lim_{x\to\infty}\frac{\sqrt x^{2}}{x} \ = \ \lim_{x\to\infty}\frac{x}{x} \ = \ 1$

$\displaystyle Note: \ \sqrt(1,000,000,000^{2}+1,000,000,000+1) \ = \ 1,000,000,000.5$

$\displaystyle \sqrt1,000,000,000^{2} \ = \ 1,000,000,000, \ hence, \ as \ x \ approaches \ infinity, \ we \ are \ left \ with \ only \ x^{2}.$