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## What is the Conjugate Method for Limits?

The **conjugate method** is used for evaluating limits that contain radicals (√). Conjugates are particularly useful in evaluating some limits because they allow us to eliminate radicals. Due to its limited scope, you should use this method *after* you have tried the direct substitution method. If the substitution method results in an indeterminate limit like 0/0, try the conjugate method to see if it works.

## What is a “Conjugate”?

A conjugate is a binomial expression with opposite middle signs (a *binomial* is the sum or the difference of two terms). The “middle sign” separates the radical part of the expression from a constant. In other words, each pair has identical terms, differing only by the sign separating them.

Don’t change all of the signs in the expression—**only the change the sign separating the radical from the constant. **This can get a bit confusing if you have multiple signs in an expression. A few examples to help clear this up:

If you have this in your formula |
…use this Conjugate |

√(x – 5) | √(x + 5) |

√(x + 5) | √(x – 5) |

√(x + 11) – 4 | √(x + 11) + 4 |

2√(2) – 1 | 2√(2) + 1 |

Using conjugates in an expression is really useful because they work to eliminate radicals. Let’s say you had a non-radical expression like 4x = 8. In order to eliminate the “4”, you would divide by 4:

- 4x/4 = 8/4 → x = 2

In the same way, you use a conjugate to eliminate a radical. That’s because the product of two conjugates containing radicals will equal a result with no radicals. For example:

- √(x – 5) * √(x + 5)
- = √x
^{2}+ 5√(x) – 5√(x) – 25 - = x – 25

## Conjugate Method for Limits: Examples

**Example problem #1:** Solve the following limit using the conjugate method:

This first example doesn’t work with substitution. Try it! Substitute in x = 5 into the expression and you’ll get an indeterminate limit (0/0).

Step 1: Multiply the numerator and the denominator by the conjugate:

Step 2: Simplify, using algebra:

Step 3: Evaluate the limit at the given point (for this example, that’s x = 5).

**Example problem #2: **Evaluate the following limit:

Step 1: Multiply the numerator and the denominator by the conjugate:

Step 2: Simplify, using algebra:

Step 3: Evaluate the limit at the given point (for this example, that’s x = 4).