Determining Limits Using Algebraic Properties

What Algebraic Properties of Limits Are For

Limits can look complicated because a single expression may contain several operations at once. Fortunately, limits behave predictably under many of the same algebraic operations you already know.

If you know the limits of two functions as , you can often determine the limit of their sum, difference, product, or quotient without doing anything more complicated than applying the appropriate property.

The key idea is simple: break a complicated limit into smaller limits, evaluate each part, and then combine the results.

The Algebraic Properties

Suppose:

Then the following properties apply, provided the required conditions are satisfied.

Sum and Difference Rules

The limit of a sum is the sum of the limits. The same idea works for subtraction.

Constant Multiple Rule

A constant can be pulled outside the limit.

Product Rule

The limit of a product is the product of the individual limits.

Quotient Rule

The quotient rule works when the limit of the denominator is not zero. If the denominator approaches zero, direct application of the quotient rule is not valid.

Power Rule

For positive integer powers, you can take the limit first and then raise the result to the power.

Root Rule

For an appropriate root, the limit can be taken inside the radical. For even roots, the expression must remain in the real-number domain near the point of interest.

Direct Substitution Often Does the Job

Algebraic properties are especially useful for polynomial and many rational-function limits because continuous functions allow direct substitution.

For example, consider:

Because a polynomial is continuous everywhere, substitute directly:

There is no need for factoring, a table, or a graph. The algebraic structure tells us exactly how the limit behaves.

Worked Examples

What Happens When Direct Substitution Gives ?

This is one of the most important distinctions in AP Calculus.

If direct substitution gives a numerical value such as , the limit is usually straightforward. But if substitution produces , you have encountered an indeterminate form.

An indeterminate form does not mean that the limit does not exist. It means that direct substitution has not given enough information to determine the limit.

For example:

Substituting produces:

At this point, the algebraic properties alone cannot finish the problem. You may need to factor, rationalize, use a special limit, or apply another method.

Using Known Limits

The algebraic properties become especially powerful when the problem gives you known limits instead of an explicit function that can simply be substituted.

Suppose you know:

Now consider:

Using the sum and constant multiple properties:

Notice what happened: we never needed to know the actual functions and . Their limits were enough.

A Reliable AP Calculus Strategy

1. Try direct substitution first. Plug the approaching value into the expression.
2. If you get a normal number, stop. For many polynomial and continuous-function limits, that is the answer.
3. If the expression contains several operations, identify the relevant limit property. Use sum, difference, constant multiple, product, quotient, power, or root rules.
4. Check the denominator. The quotient rule requires a nonzero denominator limit.
5. If substitution gives , do not conclude that the limit is zero or undefined. Recognize it as an indeterminate form and use another technique.

Common Mistakes

Practice Problems