A simple limit problem.

**guidol92** · May 23rd 2010, 07:21 PM

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?

**Prove It** · May 23rd 2010, 07:30 PM

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?

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

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

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

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

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

$= -\frac{1}{9}$ .

**mr fantastic** · May 24th 2010, 12:51 AM

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.

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