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`lim_(x->4) (x-4)/(x^3-64)`  show steps to derive the limit of `1/48`

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user7230927 | (Level 1) Salutatorian

Posted June 4, 2013 at 4:24 PM via web

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`lim_(x->4) (x-4)/(x^3-64)`  show steps to derive the limit of `1/48`

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted June 4, 2013 at 4:38 PM (Answer #1)

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You should evaluate the limit converting the difference of cubes `x^3 - 64` into a product, using the following formula, such that:

`a^3 - b^3 = (a - b)(a^2 + ab + b^2)`

Reasoning by analogy yields:

`x^3 - 64 = (x - 4)(x^2 + 4x + 16)`

Replacing `(x - 4)(x^2 + 4x + 16)` for `x^3 - 64` yields:

`lim_(x->4) (x - 4)/((x - 4)(x^2 + 4x + 16))`

Reducing duplicate factors yields:

`lim_(x->4) 1/(x^2 + 4x + 16)`

You may evaluate a limit performing the following steps, such that:

`lim_(x->a) f(x) = f(a)`

Reasoning by analogy yields:

`lim_(x->4) 1/(x^2 + 4x + 16) = 1/(4^2 + 4*4 + 16)`

`lim_(x->4) 1/(x^2 + 4x + 16) = 1/48`

Hence, evaluating the given limit yields` lim_(x->4) 1/(x^2 + 4x + 16) = 1/48.`

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