Limits and Their Properties
LimitsWe would like to the find the slope of the
tangent line to a curve…
We can’t because you need TWO points to find a slope…
Instead, we use the slope of the SECANT line because two points are available.
As the slope of the SECANT line approaches the slope of the TANGENT line, we are finding the LIMIT!
3 cases where a limit DNE…lim0
xDNE
xx
20
1limx
DNEx
0
1limsinx
DNEx
*You may not have TWO values as a limit
*Increasing without bounds
*Constantly moving between TWO points.
Limits-> Evaluated by Substitution1. Polynomials2. Radicals3. Rational Expressions…..ALL CONTINUOUS
EVERYWHERE WHEN GRAPHED
2
2
2
lim4 3
4(2) 3
4(4) 3
16 3
19
xx
If Direct Substitution Fails…1. Factor, then cancel.2. Rationalize the numerator.
Ex: Ex: 2
5
5
5
25lim
5( 5)( 5)
lim5
lim( 5)
5 5
10
x
x
x
xxx xx
x
0
0
0
0
1 1 1 1lim
1 11 1
lim( 1 1)
lim( 1 1)
1lim( 1 1)
1 12( 0 1 1)
x
x
x
x
x xx xx
x x
x
x x
x
Two Special Trig Limits…sinlim 10
xxx
1 coslim 00
xxx
0
sin 2lim0sin 2
2lim2
2(1)2
x
xxxxx
-Direct Substitution yields Undefined denominator.-Correct the limit as needed.
One Sided LimitsEvaluate from the LEFT and the RIGHTBoth limits MUST BE EQUAL in order for
the limits to exist!
25
5lim
25x
x
x
25
5lim
25x
x
x
**Both limits = 110
To Find and Asymptote1. Set the denominator equal to “zero” and
solve2. Answers are where vertical asymptotes
exist.Ex:
4( )2 5 4
2 5 4 0( 4)( 1) 04 0, 1 04, 1
f xx x
x xx xx xx x
Vertical Asymptotes @ x = 4 and x = 1.