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 $hto0$