Calculus Cookbook: Chapter 2
Download
Report
Transcript Calculus Cookbook: Chapter 2
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
x0 x
1
DNE
x0 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
x2
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 11
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…
1cos 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
x 2 5 x 4 0
( x 4)( x 1) 0
x 40, x 10
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.