Question

Question #65e7a

Answer 1

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

Explanation:

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`