## anonymous 4 years ago Evaluate the limit. As x approaches infinity, the limit of x minus the square root of x^2 minus x minus 1.

1. anonymous

$\lim_{x \rightarrow \infty}x-\sqrt{x ^{2}-x-1}$

2. anonymous

I know the answer, I just need to see how to get it. (1/2)

3. TuringTest

multiply by$\frac{x+\sqrt{x^2-x-1}}{x+\sqrt{x^2-x-1}}$

4. TuringTest

$x-\sqrt{x^2-x-1}\cdot\frac{x+\sqrt{x^2-x-1}}{x+\sqrt{x^2-x-1}}=\frac{x^2-(x^2-x-1)}{x+\sqrt{x^2-x-1}}$

5. TuringTest

$=\frac{x+1}{x+\sqrt{x^2-x-1}}=\frac{x+1}{x+x\sqrt{1-\frac1x-\frac1{x^2}}}$divide top and bottom of the fraction by x:$=\frac{1+\frac1x}{1+\sqrt{1-\frac1x-\frac1{x^2}}}$now evaluate the limit

6. anonymous

remember $(a/\infty) = 0$