Question

How do you find the limit `lim_(h->0)((3+h)^(-1)-3^-1)/h` ?

Answer 1

#lim_{h to 0}{(3+h)^{-1}-3^{-1}}/h =-1/9#

Let us simplify the quotent first.
`{(3+h)^{-1}-3^{-1}}/h`
by rewriting the -1 power as a reciprocal,
`={1/{3+h}-1/3}/h`
by taking the common denominator of the numerator,
`={{3-(3+h)}/{3(3+h)}}/h`
by cancelling out the 3's,
`={{-h}/{3(3+h)}}/h`
by cancelling out the `h`'s,
`={-1}/{3(3+h)}`

Now, let us take the limit.
#lim_{h to 0}{-1}/{3(3+h)} ={-1}/{3(3+0)} =-1/9#