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.

How Do You Compute the Limit of a Difference Quotient?

The last part of Section 10.3 asks you to compute several different difference quotients. Some of the problems ask you to compute

\underset{h \to 0}{\mathop{lim }},\frac{f(a+h)-f(a)}{h}

where f(x) and a are given to you in the problem. Here are a few examples from the board.

f(x)=4x+3 and a=1

The board above contains a mistake…do you see where this group made a mistake?

f(x)={{x}^{2}}-4 and a=1

f(x)={{x}^{2}}-1 and a=2


In the examples below, you are asked to compute a difference quotient containing x instead of a.

Compute \underset{h \to 0}{\mathop{lim }},\frac{f(x+h)-f(x)}{h} where f(x)={{x}^{2}}+2x


Compute \underset{h \to 0}{\mathop{lim }},\frac{f(x+h)-f(x)}{h} where f(x)={{x}^{2}}-x

 By the way, the correct solution to the first problem is below.

In this original calculation, f(a+h) and f(a) where switched.

How Do You Calculate a Limit Algebraically?

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.

\underset{x\to 3}{\mathop{lim }}\frac{{{x}^{2}}-5x+6}{x-3}

\underset{x\to -1}{\mathop{lim }}\frac{{{x}^{2}}-x-2}{x+1}

In the next two examples, the fractions in the numerator must be combined before the fraction may be simplified.

\underset{x\to 0}{\mathop{lim }}\frac{\frac{1}{x-6}+\frac{1}{6}}{x}

\underset{x\to 0}{\mathop{lim }}\frac{\frac{1}{4}-\frac{1}{x+4}}{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.

\underset{x\to -5}{\mathop{lim }}\frac{1}{{{\left( x+5 \right)}^{2}}}

\underset{x\to 1}{\mathop{lim }}\frac{x}{{{\left( x-1 \right)}^{2}}}

The next two examples show how to rationalize the numerator to do a limit.

\underset{x\to 1}{\mathop{lim }}\frac{\sqrt{x}-1}{x-1}

This example may be done two different ways as the next two boards demonstrate.

\underset{x\to 4}{\mathop{lim }}\frac{\sqrt{x}-2}{x-4}

How Do You Do Limits with Square Roots?

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 \displaystyle \frac{\sqrt{x}-1}{x-1}. In the handout below, we’ll look at limits involving this fraction.

Other limits that have square roots can be handled with similar strategies.