lim h->0 for ((5+h)^-1 - 5^-1)/h

Question

lim h->0 for ((5+h)^-1 - 5^-1)/h

Answer 1

If f(x) = 1/x, then the given limit is the definition of f'(5).

 

f'(x) = -1/x²

 

So, the value of the given limit is f'(5) = -1/25.

Answer 2

Rewrite so that you have no negative exponents.

 

lim      [1 / (5 + h)] - [1 / 5]

h->0  ____________________

                         h

 

 

As you can see, you are taking the limit of  f(x)=1/x   at    x=5. 

 

 

The write using the LCD of the numerator.

 

 

lim              5 - (5 + h)                1

h->0     ________________ *  _____

                    5(5 + h)                  h

 

 

 

lim                - h

h->0   ______________

                  5h(5 + h)

 

 

Cancel out h terms.

 

lim              - 1

h->0     ____________

                5(5 + h)

 

 

Now we can plug in h=0.

 

lim          -1

h->0    _____

               25

 

 

The limit is  -1/25.