A simple limit problem.

Jan 2010
11
1
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
Aug 2008
12,883
4,999
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
Dec 2007
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6,768
Zeitgeist
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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