DO NOW!
Titlepp. 58:
HW
•Estimate
Loremnumerically:
ipsum dolor sit amet, consectetuer
adipiscing elit. Vivamus et magna.
Fusce sed
Objective
x  27
1) sem
lim sed magna suscipit egestas.
Solve limits analytically by:
x 3
• Lorem ipsum dolor sit amet,
consectetuer
• Direct
substitution;
adipiscing elit. Vivamus et• magna.
Fusce
sed
Dividing out; and
x 1 2
lim sed magna suscipit egestas.
2) sem
3
x3
x5
x5
• Rationalizing the
numerator.
Solving Limits Analytically
Limit Laws, Direct Substitution,
Dividing out, and rationalizing
the numerator
Methods of Finding Limits
• Numerically = in a table
• Graphically = on a graph
• Analytically = using algebra/equations
Solving Limits Analytically
• Limit Laws
• Direct Substitution
• Dividing Out
• Rationalizing the Numerator
Limit Laws
Let b, c  , n   > 0, and
lim f ( x )  L
xc
lim g ( x )  K
xc
Then:
1. lim 𝑏 ∙ 𝑓 𝑥 = 𝑏𝐿
𝑥→𝑐
lim  f ( x )  g ( x )   L  K
2. x  c
3. lim 𝑓 𝑥 ∙ 𝑔 𝑥
𝑥→𝑐
=𝐿∙𝐾
Limit Laws
Let b, c  , n   > 0, and
lim f ( x )  L
xc
Then:
4. lim
xc
5.
lim g ( x )  K
xc
 f (x) 
L

 
 g(x) K
n
n
lim  f ( x )   L
xc
Direct Substitution
This method of solution can only be used when
the functions are:
– Polynomial functions
– Rational functions with nonzero denominators
– Positive Integer Radical functions (square, cube,
quartic, etc roots)
– Trigonometric functions
– Composite functions of above
Example
Find the limit for each function.
x  4x  1
2
lim
x 1
x2

3
lim  s in x
x
2

When the limit gives
an indeterminant form…
Dividing Out
When you have a rational function where you can “divide out” a factor.
x  64
2
lim
x  8
x8
Rationalizing the Numerator
When you have a rational function where the numerator
has a sum or difference and a radical.
lim
𝑥→3
𝑥−2−1
𝑥−3
Classwork
Find the limit of each function, analytically.
x  3x  4
2
lim
x  1
lim
x3
5x  5
1
4x
x3
lim
2  x 
x0
lim
x  4
2
4
x
x  4
4
2
20  x
DO NOW!
Titlepp. 66:
HW
67-80 all
each ipsum
limit: dolor sit amet, consectetuer
•Find
Lorem
adipiscing elit. Vivamus et magna.
Fusce sed
Objective
sem sed
suscipit egestas.
𝒙 + 𝟒magna
−𝟒
Solve limits analytically by:
1) lim
𝒙 −𝟏𝟐
𝒙 → 𝟏𝟐
• Lorem
ipsum dolor sit amet,
consectetuer
• Simplifying
adipiscing elit. Vivamus et• magna.
Fusce
sed
Direct Substitution
sem sed magna suscipit egestas.
2)
𝒙 + 𝒉 𝟒 − 𝒙𝟒
lim
𝒉
ℎ →𝟎
• Rationalizing the Numerator
• Trig rules
Trig Limits
Sometimes the limit of a trig function is indeterminant.
In this case, remember these:
sin 𝑥
=1
𝑥→0 𝑥
𝒂. ) lim
cos 𝑥 − 1
𝒃. ) lim
=0
𝑥→0
𝑥
sin 𝑥
𝑥
𝒄. ) lim
= lim
=1
𝑥→0 𝑥
𝑥→0 sin 𝑥
Trig Limits
Use limit laws, trig props, and old concepts from algebra.
sin 𝑥
𝑥→0 5𝑥
1.) lim
sin 2𝑥
𝑥→0 2𝑥
2.) lim
Trig Limits
Use limit laws, trig props, and old concepts from algebra.
sin 5𝑥
𝑥→0 𝑥
3.) lim
1−cos 𝑥
𝑥→0 4𝑥
4.) lim
Trig Limits
Use limit laws, trig props, and old concepts from algebra.
sin 2𝑥
𝑥→0 7𝑥
5.) lim
sin 3𝑥
𝑥→0 sin 4𝑥
6.) lim
Trig Limits
Use limit laws, trig props, and old concepts from algebra.
tan 𝑥
𝑥→0 7𝑥
7.) lim
Classwork
Complete the joke worksheets provided.
HW: p. 66 from worksheet: 67-80 all.