I have a calculus review test and our teacher gave us a problem I can't crack

Question

(1/(a+h) - 1/a)/h

Answer 1

It looks like you are trying to find the limit as h → 0 of the expression.  It should give -1/a², which is the derivative of f(a) = 1/a.

 

Look at the first term in the parenthesis.

 

1/(a + h) = 1 / [a(1 + h/a)] ≅ (1/a)(1 - h/a) for very small h/a 

 

This used the approximation 

 

1/(1 + x) ≅ 1 - x for x << 1

 

Plug this into the original expression and get

 

limit _h→0 [1/(a + h) - 1/a]/h = limit _h→0 [(1/a)(1 - h/a) - 1/a]/h = [-h/a²]/h = -1/a²

 

 

 

Answer 2

(1/(a+h) - 1/a)/h add the two fractions using the common denominator a(a+h)

a/(a(a+h))-(a+h)/(a(a+h))=(a-(a+h))/(a(a+h))=-h/(a(a+h)) Now divide it by h to get

-1/(a(a+h))