Last class we learned that the limit of as approaches does not depend on the value of at It may happen however that the limit is precisely In such cases the limit can be evaluated by ID: 208810
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Evaluating Limits AnalyticallySlide2
Last class, we learned that the limit of
as
approaches does not depend on the value of at It may happen, however, that the limit is precisely In such cases, the limit can be evaluated by direct substitution.
Slide3
THEOREM 2.1
Some Basic Limits
Let be real numbers and let be a positive integer. Then, 1.
2.
3.Slide4
Properties of Limits HandoutSlide5
Ex 3. Let:
Find:
a)
b)
c)
d)Slide6
So, if it’s a polynomial or rational with nonzero denominator, just plug it in…Slide7
In fact,
always
try to plug in and see what happens…If you get a number back, then great! That’s the limit!If not… Slide8
DIVIDING OUT (FACTORING) TECHNIQUE
Ex 4.
Slide9
COMMON DENOMINATOR TECHNIQUE
Ex 5.
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RATIONALIZING (CONJUGATE) TECHNIQUE
Ex 6.
Slide11
THEOREM 2.9
Two Special Limits
1. 2.
Slide12
Ex 7.
Slide13
Ex 8.
Slide14
Ex 9.
Slide15
So, in summary:
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!