EVALUATING LIMITS ANALYTICALLY
Last class, we learned that the limit of 𝑓(𝑥) as 𝑥 approaches 𝑎 does not depend on
the value of 𝑓 at 𝑥 = 𝑎.
It may happen, however, that the limit is precisely 𝑓 𝑎 , which means
lim 𝑓 𝑥 = 𝑓(𝑎)
𝑥→𝑎
In such cases, the limit can be evaluated by direct substitution.
THEOREM 2.1 Some Basic Limits
Let 𝑎 and 𝑏 be real numbers and let 𝑛 be a positive integer. Then,
1.
lim 𝑏 = 𝑏
𝑥→𝑎
2.
lim 𝑥 = 𝑎
𝑥→𝑎
3.
lim 𝑥 𝑛 = 𝑎𝑛
𝑥→𝑎
Properties of Limits Handout
Ex 3. Let:
lim 𝑓 𝑥 = 4
𝑥→𝑎
Find:
lim [𝑓 𝑥 + ℎ 𝑥 ]
a) 𝑥→𝑎
𝑓 𝑥
b) lim
𝑥→𝑎 ℎ 𝑥
lim 𝑔 𝑥 = 0
lim ℎ 𝑥 = −3
𝑥→𝑎
𝑥→𝑎
c)
d)
lim ℎ 𝑥
2
𝑥→𝑎
ℎ 𝑥
lim
𝑥→𝑎 𝑔 𝑥
So, if it’s a polynomial or rational with
nonzero denominator, just plug it in…
In fact, always try to plug in 𝑎 and see what happens…
If you get a number back, then great! That’s the limit!
If not…
DIVIDING OUT (FACTORING) TECHNIQUE
𝑥2 + 𝑥 − 6
Ex 4. lim
𝑥→−3
𝑥+3
COMMON DENOMINATOR TECHNIQUE
Ex 5.
1
1
−
lim 𝑥 + 4 4
𝑥→0
𝑥
RATIONALIZING (CONJUGATE) TECHNIQUE
Ex 6.
𝑥+1−1
lim
𝑥→0
𝑥
THEOREM 2.9 Two Special Limits
1.
sin 𝑥
lim
=1
𝑥→0 𝑥
2.
1 − cos 𝑥
lim
=0
𝑥→0
𝑥
Ex 7.
tan 𝑥
lim
𝑥→0 𝑥
Ex 8.
sin 4𝑥
lim
𝑥→0 5𝑥
Ex 9.
2−𝑥
lim+
𝑥→2 |2 − 𝑥|
So, in summary:
1.
2.
3.
4.
5.
Try Direct Substitution! You might get lucky! If not…
Factoring (Dividing Out)
Common Denominator (if you see fractions in fractions)
Rationalizing (if you see a radical)
Remember special trig limits!