HELP find this limit: lim [x→∞] {x (sqrt[x^2 + 1] - x)}

[spartas]

\[\lim_{x \rightarrow \ \infty}x(\sqrt{x^2+1}-x)\]

and does it have a different value if :
\[\lim_{x \rightarrow \ -\infty\ }x(\sqrt{x^2+1}-x)\]

 

[ksdhart2]

What are your thoughts? What have you tried? Please share with us any and all work you've done on this problem, even if you know it's wrong. If you're stuck at the very beginning, a good first step might be to recall back to the algebra you learned so long ago, and note:

\(\displaystyle x\left(\sqrt{x^2+1}-x\right)=x\sqrt{x^2+1}-x^2\)

 

[spartas]

i tried solving it leads to 0, but in the answers it has to be 1/2
and the second one is \[-\infty\]

 

[HallsofIvy]

i tried solving it leads to 0, but in the answers it has to be 1/2
and the second one is \[-\infty\]

You aren't telling us anything! HOW did you try solving it?

When you have a square root, an almost obvious thing to try is to get rid of the square root by using the fact that \(\displaystyle (a- b)(a+ b)= a^2- b^2\). Here, try multiplying \(\displaystyle \frac{x\sqrt{x^2+ 1}- x^2}{1}\) by \(\displaystyle \frac{x\sqrt{x^2+ 1}+ x^2}{x\sqrt{x^2+ 1}+ x^2}\). What does that give?