CK-12-Pre-Calculus Concepts

14.5. Rationalization to Find Limits http://www.ck12.org

14.5 Rationalization to Find Limits

Here you will evaluate limits analytically using rationalization.
Some limits cannot be evaluated directly by substitution and no factors immediately cancel. In these situations there
is another algebraic technique to try called rationalization. With rationalization, you make the numerator and the
denominator of an expression rational by using the properties of conjugate pairs.
How do you evaluate the following limit using rationalization?

xlim→ 16

√x− 4

x− 16

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MEDIA

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http://www.youtube.com/watch?v=ouWAhqeAaik James Sousa: Find a Limit Requiring Rationalizing

Guidance

The properties of conjugates are used in a variety of places in PreCalculus.
Conjugates can be used to simplify expressions with a radical in the denominator:

5

1 +√ 3 =

(^5

1 +√ 3 )·

( 1 −√ 3 )

( 1 −√ 3 )=^5 −^5

√ 3

1 − 3 =^5 −^5

√ 3

− 2

Conjugates can be used to simplify complex numbers withiin the denominator:

2 +^43 i=( 2 +^43 i)·((^22 −−^33 ii))=^84 −+^129 i=^8 − 1312 i
Here, they can be used to transform an expression in a limit problem that does not immediately factor to one that
does immediately factor.

xlim→ 16

(√x− 4 )

(x− 16 ) ·

(√x+ 4 )

(√x+ 4 )=xlim→ 16 (x− 16 (x)(−^16 √)x+ 4 )

Now you can cancel the common factors in the numerator and denominator and use substitution to finish evaluating
the limit.
The rationalizing technique works because when you algebraically manipulate the expression in the limit to an
equivalent expression, the resulting limit will be the same. Sometimes you must do a variety of different algebraic
manipulations in order avoid a zero in the denominator when using the substitution method.
Example A
Evaluate the following limit: limx→ 3 √xx^2 −−√^93.