Question

Prove without L'Hospital's rule

Lim x to inf (x-(x^2+1)^(1/2))

Answer 1

\[\lim_{x \rightarrow \infty} \left( x-\sqrt{x^{2}+1}\right) = 0\]

Answer 2

\[\lim_{x \rightarrow \infty} \left( x-\sqrt{x^{2}+1}\right) \]\[=\lim_{x \rightarrow \infty} x-\sqrt{x^{2}+1}\times \frac{x+\sqrt{x^{2}+1}}{x+\sqrt{x^{2}+1}}\]\[=\lim_{x \rightarrow \infty}\frac{x^2-(x^{2}+1)}{x+\sqrt{x^{2}+1}}\]\[=\lim_{x \rightarrow \infty}\frac{-1}{x+\sqrt{x^{2}+1}}\]\[=\lim_{x \rightarrow \infty}\frac{\frac{-1}{x}}{\frac{x+\sqrt{x^{2}+1}}{x}}\]\[=\lim_{x \rightarrow \infty}\frac{\frac{-1}{x}}{1+\sqrt{1+\frac{1}{x^2}}}\]I guess you can do it?

Answer 3

great thx a lot!

Answer 4

You're welcome :)