# A simple limit problem.

#### 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?

#### Prove It

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

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

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

#### mr fantastic

MHF Hall of Fame
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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