Question

Evaluate the limit. As x approaches infinity, the limit of x minus the square root of x^2 minus x minus 1.

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

remember \[(a/\infty) = 0\]