# Infinite Limit Question

#### kaiser0792

Can anyone please explain to me how $$\displaystyle \lim_{n->\infty}\left| \frac{nx}{n+1} \right| = |x|$$? Does it not require L' Hopital's Rule and if so, how does it evaluate to |x|?

#### TheEmptySet

MHF Hall of Honor
Can anyone please explain to me how $$\displaystyle \lim_{n->\infty}\left| \frac{nx}{n+1} \right| = |x|$$? Does it not require L' Hopital's Rule and if so, how does it evaluate to |x|?
$$\displaystyle \left| \frac{nx}{n+1} \right|=\bigg|\frac{n}{n} \bigg|\left| \frac{x}{1+\frac{1}{n}} \right| = \left| \frac{x}{1+\frac{1}{n}} \right|$$

HallsofIvy

Nice, thank you!

#### matheagle

MHF Hall of Honor
I'd pull out the |x| since n is certainly positive.

$$\displaystyle \left | \frac{nx}{n+1} \right| = |x|\left({n\over n+1}\right)\to |x|\cdot 1=|x|$$