Problem Evaluate the difference quotient for f (x) = x^{2} – 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.

You can recognize the limits by what happens when you substitute the value x approaches into the expression. If it gives 0/0, there is algebra that you can do to find the exact value of the limit.

In the first two examples, the expression may be factored and simplified…then you can substitute the value for x.

The next two examples are designed to throw you off. When you substitute the value into the expression, you do not get 0/0. This means you need to use a table or graph to get the limit.

In section 1.3, you’ll be learning how to compute limits using algebra. There are several examples that involve factoring and fractions, but none with square roots. There are several ways to simplify expressions like $latex \displaystyle \frac{\sqrt{x}-1}{x-1}$. In the handout below, we’ll look at limits involving this fraction.