# Thread: Need some help Finding a limit (if it exists even)

1. ## Need some help Finding a limit (if it exists even)

Howdy,

I need some help finding this limit:

$lim \Delta x -> 0$

$\sqrt{x+\Delta x}-\sqrt{x}$
______________________________

$\Delta x$

(P.S. Sorry I was having a bit of trouble using the \frac command, just so you know it's suppose to be delta X on the bottom of the fraction as I tried to show using the ___). Thanks.

jimmyp

2. Originally Posted by jimmyp
Howdy,

I need some help finding this limit:

$lim \Delta x -> 0$

$\sqrt{x+\Delta x}-\sqrt{x}$
______________________________

$\Delta x$

(P.S. Sorry I was having a bit of trouble using the \frac command, just so you know it's suppose to be delta X on the bottom of the fraction as I tried to show using the ___). Thanks.

jimmyp
first we need to multiply the numerator and denominator by the conjugate of the numerator

$\lim_{\Delta x \to 0}\frac{\sqrt{x+\Delta x}-\sqrt{x}}{\Delta x}\left( \frac{\sqrt{x+\Delta x}+\sqrt{x}}{\sqrt{x+\Delta x}+\sqrt{x}}\right)$

Simplifying we get

$\lim_{\Delta x \to 0}\frac{\Delta x}{\Delta x(\sqrt{x+\Delta x}+\sqrt{x})}=\lim_{\Delta x \to 0}\frac{1}{(\sqrt{x+\Delta x}+\sqrt{x})}$

Finally taking the limit we get

$\frac{1}{2\sqrt{x}}$

3. Originally Posted by TheEmptySet
first we need to multiply the numerator and denominator by the conjugate of the numerator....
That "multiplying by the conjugate" thing, by the way, is a "trick" you should remember. You will almost certainly need it on the next test!