14.5. Rationalization to Find Limits http://www.ck12.org
xlim→ 16√x− 4
x− 16 =xlim→ 16(√x− 4 )
(x− 16 )·(√x+ 4 )
(√x+ 4 )
=xlim→ 16 (x− 16 (x−)(^16 √)x+ 4 )
=xlim→ 16 (√x+ 4 )
= 4 + 4
= 8Vocabulary
Rationalizationgenerally means to multiply a rational function by a clever form of one in order to eliminate radical
symbols or imaginary numbers in the denominator.Rationalizationis also a technique used to evaluate limits in
order to avoid having a zero in the denominator when you substitute.
Guided Practice
- Evaluate the following limit: limx→ 0 (^2 +x)−x^1 −^2 −^1.
- Evaluate the following limit: limx→− 3
√
x^2 x+− 35 − (^2).
- Evaluate the following limit: limx→ 0
(
3
x√ 9 −x−(^1) x
)
.
Answers:
limx→ 0 (^2 +x)− (^1) − 2 − 1
x =limx→^0
x+^12 −^12
x ·
(x+ 2 )· 2
(x+ 2 )· 2
=limx→ 022 −x((xx++ 22 ))
=limx→ 02 x(−x+x 2 )
=limx→ 02 (x−+^12 )
=− 2 ( 01 + 2 )
=−^14