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What is limit f(x)-f(0)/x if x ---->0? f (x)=1/(x+1)(x+2)

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luvgoj | Student, Undergraduate

Posted August 23, 2013 at 11:57 AM via web

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What is limit f(x)-f(0)/x if x ---->0?

f (x)=1/(x+1)(x+2)

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted August 23, 2013 at 12:11 PM (Answer #1)

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You need to use the limit definition of derivative, such that:

`f'(c) = lim_(x->c) (f(x) - f(c))/(x - c)`

Since the problem requests for you to evaluate the limit `lim_(x->0) (f(x) - f(0))/(x - 0)` , you need to evaluate the derivative of the function at `x = 0` .

You need to differentiate the function with respect to x, using the quotient rule, such that:

`f'(x) = (1'*(x+1)(x+2) - 1*((x+1)(x+2))')/((x+1)^2(x+2)^2)`

`f'(x) = (-(x^2 + 3x + 2)')/((x+1)^2(x+2)^2)`

`f'(x) =(-2x - 3)/((x+1)^2(x+2)^2)`

You need to evaluate `f'(0)` , hence, you need to replace 0 for x, such that:

`f'(0) = (-2*0 - 3)/((0+1)^2(0+2)^2)`

`f'(0) = -3/4`

Hence, evaluating the given limit using the limit definition of derivative, yields `lim_(x->0) (f(x) - f(0))/(x - 0) = -3/4.`

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