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!
Best Answer
Let $t^2=x+25$, then $t=\sqrt{x+25}.$ Then we have $$\lim_{t\to 5}\frac{5-t}{t^2-25}=\lim_{t\to 5}\dfrac{5-t}{(t+5)(t-5)}=-\lim_{t\to 5}\dfrac{5-t}{(t+5)(5-t)}=-\lim_{t\to 5}\dfrac{1}{t+5}=-\dfrac{1}{10}.$$