Limits and Their Properties
Limits
 We 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…
x
lim  DNE
x0 x
1
 DNE
x0 x 2
lim
1
lim sin  DNE
x 0
x
*You may not have TWO
values as a limit
*Increasing without bounds
*Constantly moving
between TWO points.
Limits-> Evaluated by Substitution
 1. Polynomials
 2. Radicals
 3. Rational Expressions…..ALL CONTINUOUS
EVERYWHERE WHEN GRAPHED
lim 4 x 2  3
x2
 4(2) 2  3
 4(4)  3
16  3
19
If Direct Substitution Fails…
 1. Factor, then cancel.
 2. Rationalize the numerator.
 Ex:
Ex:
x  25
lim
x 5 x 5
( x 5)( x 5)
lim
x 5
x 5
lim( x 5)
2
x 5
5  5
10
x 1 1
x 1 1

x 0
x
x 1 1
x 11
lim
x  0 x(
x 1 1)
x
lim
x  0 x(
x 1 1)
1
lim
x 0 (
x 1 1)
1
1


( 0 1 1) 2
lim
Two Special Trig Limits…
1cos x
sin x
lim
0
lim
1
x0 x
x0 x
sin 2 x
lim
x 0 x
sin 2 x
2 lim
x 0 2 x
 2(1)
2
-Direct Substitution yields
Undefined denominator.
-Correct the limit as needed.
Continuity
 A graph is continuous if…
 1. No gaps
 2. No holes
 3. No jumps
One Sided Limits
 Evaluate from the LEFT and the RIGHT
 Both limits MUST BE EQUAL in order for the limits to
exist!
x 5
lim 2
x 5 x  25
**Both limits =
x 5
lim 2
x 5 x  25
1
10
Infinite Limits
 A limit in which f(x) increases or decreases without
bound as “x” approaches “c”.
To Find and Asymptote
 1. Set the denominator equal to “zero” and solve
 2. Answers are where vertical asymptotes exist.
 Ex:
f ( x) 
4
2
x 5 x  4
2
x 5 x  4  0
( x  4)( x 1) 0
x  40, x 10
x  4, x  1
Vertical Asymptotes
@ x = 4 and x = 1.
To Find Infinite Limits
 1. Factor numerator and/or denominator if possible.
 2. Cancel, if possible.
 3. With what remains:
 A. Set numerator equal to zero to find x- intercepts.
 B.. Set denominator equal to zero to find vertical
asymptotes
 4. Select appropriate points to find designated limits.