Evaluate the difference quotient?

Question

Evaluate the difference quotient f(x) -f(a)/ x-a for f(x)= 1/x2. Thank you very much!

Answer 1

Let s = [ f(x) - f(a) ] / (x - a).

s = [ 1 / x^2 - 1 / a^2 ] / (x - a)

= [ (a^2 - x^2) / (a^2 x^2) ] / (x - a)

= (a - x)(a + x) / [ (x - a) (a^2 x^2) ]

= - (x + a)(a^2 x^2).

Lim (x - >  a) [ s ] = - 2a / a^4

= - 2 / a^3.