Question #65e7a

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Answer 1 · 2016-11-06T00:26:09

Please see the explanation for the steps leading to the answer:
#(-1)/((x + h - 8)(x - 8))#

GIven: #f(x) = 1/(x - 8)#

Then #f(x + h) = 1/(x + h - 8)#

Substituting into the difference quotient:

#{1/(x + h - 8) - 1/(x - 8)}/h#

Multiply by 1 in the form #((x + h - 8)(x - 8))/((x + h - 8)(x - 8))#

#{1/(x + h - 8) - 1/(x - 8)}/h((x + h - 8)(x - 8))/((x + h - 8)(x - 8))#

Multiply by the 1s in the upper numerators and h in the lower denominator:

#{((x + h - 8)(x - 8))/(x + h - 8) - ((x + h - 8)(x - 8))/(x - 8)}/(h(x + h - 8)(x - 8))#

Now, I will show you which factors cancel in the upper part:

#{(cancel((x + h - 8))(x - 8))/cancel((x + h - 8)) - ((x + h - 8)cancel((x - 8)))/cancel((x - 8))}/(h(x + h - 8)(x - 8))#

Remove the canceled factors:

#{(x - 8) - (x + h - 8)}/(h(x + h - 8)(x - 8))#

Distribute the implicit -1:

#{x - 8 - x - h + 8}/(h(x + h - 8)(x - 8))#

When we combine like terms in the numerator, we are left with -h:

#(-h)/(h(x + h - 8)(x - 8))#

#-h/h# reduces to -1:

#(-1)/((x + h - 8)(x - 8))#

Now it will be safe for you to let #hto0#