TITLE: Solving limit without L'Hôpital
QUESTION [11 upvotes]: I need to solve this limit without L'Hôpital's rule. These questions always seem to have some algebraic trick which I just can't see this time.
$$ \lim_{x\to0} \frac{5-\sqrt{x+25}}{x}$$
Could someone give me a hint as to what I need to do to the fraction to make this work? Thanks!

REPLY [4 votes]: $$\lim_{x\to0}\frac{5-\sqrt{x+25}}{x}=\lim_{x\to0}\frac{5-\sqrt{x+25}}{(\sqrt{x+25})²-25}=\lim_{x\to0}\frac{5-\sqrt{x+25}}{-(5-\sqrt{x+25})(5+\sqrt{x+25})}=\lim_{x\to0}\frac{-1}{5+\sqrt{x+25}}=\frac{-1}{10}$$