# A simple limit problem.

• May 23rd 2010, 07:21 PM
guidol92
A simple limit problem.
Hello.
I have to find the limit as x approaches 0 of the function
[ [(1/(3+x)] -(1/3) ] / x i tried direct substitution and to rationalize the denominator and numerator, but neither works...

what shoul i do?
• May 23rd 2010, 07:30 PM
Prove It
Quote:

Originally Posted by guidol92
Hello.
I have to find the limit as x approaches 0 of the function
[ [(1/(3+x)] -(1/3) ] / x i tried direct substitution and to rationalize the denominator and numerator, but neither works...

what shoul i do?

$\displaystyle \frac{\frac{1}{3 + x} - \frac{1}{3}}{x} = \frac{\frac{3 - (3 + x)}{3(3 + x)}}{x}$

$\displaystyle = \frac{-\frac{x}{3(3 + x)}}{x}$

$\displaystyle = -\frac{1}{3(3 + x)}$.

So $\displaystyle \lim_{x \to 0}\left(\frac{\frac{1}{3 + x} - \frac{1}{3}}{x}\right) = \lim_{x \to 0}\left[-\frac{1}{3(3 + x)}\right]$

$\displaystyle = -\frac{1}{3(3)}$

$\displaystyle = -\frac{1}{9}$.
• May 24th 2010, 12:51 AM
mr fantastic
Quote:

Originally Posted by guidol92
Hello.
I have to find the limit as x approaches 0 of the function
[ [(1/(3+x)] -(1/3) ] / x i tried direct substitution and to rationalize the denominator and numerator, but neither works...

what shoul i do?

It can also be recognised as having the same form as the derivative from first principles of f(t) = 1/t evaluated at t = 3.