14.4. Substitution to Find Limits http://www.ck12.org
xlim→ 2 x(^2) − 4
x− 2 =limx→^2
(x− 2 )(x+ 2 )
(x− 2 )
=limx→ 2 (x+ 2 )
= 2 + 2
= 4
Example C
Evaluate the following limit analytically: limx→ 4 x^2 −x−x− 412
Solution:
xlim→ 4 x
(^2) −x− 12
x− 4 =limx→ 4
(x− 4 )(x+ 3 )
(x− 4 )
=limx→ 4 (x+ 3 )
= 4 + 3
= 7
Concept Problem Revisited
In order to decide whether substitution is an appropriate first step you can always just try it. You’ll know it won’t
work if you end up trying to evaluate an expression with a denominator equal to zero. If this happens, go back and
try to factor and cancel, and then try substituting again.
Vocabulary
Substitutionis a method of determining limits where the value thatxis approaching is substituted into the function
and the result is evaluated. This is one way to evaluate a limitanalytically.
Guided Practice
- Evaluate the following limit analytically.
limx→ 3 xx^2 −− 39 - Evaluate the following limit analytically.
limt→ 4
√
t+ 32- Evaluate the following limit analytically.
limy→ 43 |yy+− 41 |
Answers:
- limx→ 3 xx^2 −− 39 =limx→ 3 (x−(^3 t−)(x 3 +)^3 )=limx→ 3 (x+ 3 ) = 6
- limt→ 4
√
t+ 32 =√
4 + 32 =
√
36 = 6
- limy→ 43 |yy+− 41 |=^3 | 44 +− 41 |=^38 ·^3 =^98