How Do You Evaluate The Limit Of A Difference Quotient?

Problem Evaluate the difference quotient  for f (x) = x² – 2x + 4.

This is a little different from   but works the same way. Since a value is not supplied for x, we just leave it and work out the limit. Start by evaluating  f (x + h):

Make sure you FOIL the square out and distribute the negative.

Now put this along with f (x)  into the difference quotient.

As h gets smaller and smaller, the term in the middle gets smaller. This means the limit is equal to 2x – 2. Since the other terms do not contain x, they are unaffected when h gets small.