Find a function f and a number a so that the following limit is the derivative of f at a:

Question

Find a function f and a number a so that the following limit is the derivative of f at a:

 

lim (h --> 0)  ((1/(9 + h)) - 1/3) / h

Answer 1

By the definition of the derivative:

 

lim(h→0) [(f(h+a) - f(a))/h]

 

From above:

f(a) = 1/3

f(a+h) = 1/(9+h)

 

I'd venture to guess that the function cannot be a derivative, since the numerator does not equal zero when h approaches 0. (Unless you forgot a square root on the f(a + h) expression)

 

IF: the question was for

lim(h→0) [(√(1/(9+h)) - 1/3)/h]

 

Then, the function is:

 

f(x) = √(1/x)

a = 9

f(a) = 1/3