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What is limit x^4-16/x-2, x go to 2 ? no hospital

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mccee | Student, Undergraduate | eNoter

Posted May 7, 2012 at 4:25 PM via web

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What is limit x^4-16/x-2, x go to 2 ? no hospital

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted May 7, 2012 at 4:38 PM (Answer #1)

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You should substitute 2 for x in `(x^4-16)/(x-2)`  such that:

`lim_(x-gt2) (x^4-16)/(x-2) = (16-16)/(2-2) = 0/0`

Since the limit is indeterminate and you should not use l'Hospital's theorem, then you should evaluate the limit using the formula of difference of squares such that:

`lim_(x-gt2) (x^4-16)/(x-2) = lim_(x-gt2) ((x^2 - 4)(x^2 + 4))/(x - 2)`

You may use again the formula of difference of squares such that:

`lim_(x-gt2) ((x^2 - 4)(x^2 + 4))/(x - 2) =lim_(x-gt2) ((x-2)(x+2)(x^2 + 4))/(x - 2)`

You should reduce like terms such that:

`lim_(x-gt2) ((x-2)(x+2)(x^2 + 4))/(x - 2) = lim_(x-gt2) (x+2)(x^2 + 4)`

You should substitute 2 for x in `lim_(x-gt2) (x+2)(x^2 + 4)`  such that:

`lim_(x-gt2) (x+2)(x^2 + 4) = (2+2)(4 + 4) = 32`

Hence, evaluating the limit yields `lim_(x-gt2) (x^4-16)/(x-2) = 32.`

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