Evaluate the limit

Question

lim (h --> 0) ((3 + h)^-1 - 3^-1)/h

 

 

ANSWER: - 1/9

Chap 2, pp13

Answer 1

This limit is the definition of the derivative of f(x) = 1/x evaluated at        x = 3.

 

If f(x) = 1/x, then f'(x) = -1/x².

 

So, the value of the limit is f'(3) = -1/9.

Answer 2

Hi Mia,

 

The way to evaluate this limit is to change the way it looks before substituting 0 for h.

 

Here is how to rewrite the expression (but I am not going to show the lim as x ->0).

 

(rewrite with positive exponent, them multiply top and bottom by LCD)

 

1/(3+h) - 1/3        3(3+h)

----------------- .   ----------------- 

       h                   3(3+h)

 

That gives: 3 - (3+h)                 -h                    -1

                 ---------------  =   -------------- = ------------

                  3h(3+h)                h(9+3h)         (9+3h)

 

Now the lim as h->0 is:

 

-1                    -1

----------- =    ------

(9+3(0))            9

 

L'Hospital's rule does not apply here because the function as written originally is not one of the indefinite forms:

 

0/0 or ∞/∞.

 

 

Answer 3

Use L'Hospital Rule.  Derivative of numerator divided by derivative of denominator.

 

lim         -(3 + h)^-2 / 1

h-->0

 

 

lim       -3^-2

h-->0

 

 

lim     -(1 / 9)

h-->0

 

 

As h approaches zero, the limit is -1/9.