Download Limits and Continuity

AP Calculus AB/BC
Unit 1: Limits and Continuity
Target 1.1 The student will be able to describe the concept of a limit (both one-sided and two-sided) in
his/her own words, and explain how a limit can fail to exist.
Target 1.2 The student will be able to evaluate a limit using the following approaches: algebraically
(including direct substitution and indeterminate form), graphically, and numerically (from a table of values).
The y -value that a graph is approaching as x approaches a specific value is the LIMIT of the function. The
graph may or may not actually contain the point it is approaching. lim f ( x)
x→a
For a limit to exist, both sides of the graph must be approaching the same y -value at that x -coordinate.
lim f ( x) means the limit as f (x) approaches the x -value from the left.
x→a −
lim f ( x) means the limit as f (x) approaches the x -value from the right.
x→a +
When a limit evaluates as lim f ( x ) =
x→a
0
the answer is considered indeterminate form, which is not an actual
0
answer. The problem must be completed through other means, such as factoring and simplifying, rationalizing
the numerator, simplifying a complex fraction, etc., and then re-evaluated at the limit value x = a .
Special trigonometric limit rules:
sin x
1. lim
=1
x →0
x
cos x − 1
=0
x →0
x
2. lim
The Greatest Integer Function
f ( x ) = x or f ( x) = int(x) refers to the greatest integer function. The
graph is shown at the right. This function evaluates the value of x to be the
greatest integer less than or equal to the value of x .
AP Calculus AB/BC
Unit 1: Limits and Continuity
Target 1.3 The student will be able to evaluate limits involving infinity.
Target 1.4 The student will be able to define vertical and horizontal asymptotes and end-behavior using
limits.
The line y = b is a horizontal asymptote of a function if either lim f ( x) = b or lim f ( x) = b .
x→∞
x→−∞
Evaluating Limits at Infinity
f ( x)
where f (x) and g (x) are polynomial functions, compare the degrees (highest
g ( x)
exponents) of each to evaluate the limit…
When evaluating lim
x →∞
f ( x)
= ∞ if the degree of f (x) is greater than the degree of g (x) .
g ( x)
•
lim
•
f ( x) a
= if the degree of f (x) is equal to the degree of g (x) ,
g ( x) b
where a is the lead coefficient of f (x) and b is the lead coefficient of g (x) .
•
lim
x→∞
lim
x→∞
x →∞
f ( x)
= 0 if the degree of f (x) is less than the degree of g (x) .
g ( x)
The line x = a is a vertical asymptote of a function if either lim+ f ( x) = ±∞ or lim− f ( x) = ±∞
x→a
x→a
End behavior models can be found by focusing on the lead terms in both the numerator and denominator of a
rational function.
AP Calculus AB/BC
Unit 1: Limits and Continuity
Target 1.5 The student will be able to define continuity at a point, and describe common discontinuities in a
function.
Target 1.6 The student will be able to explain and apply the Intermediate Value Theorem.
A graph is continuous if you can trace it without picking up your pencil.
A function is continuous at a point c, if lim f ( x) = f (c) . In expanded form, lim− f ( x) = lim+ f ( x) = f (c) .
x→c
x →c
x→c
Types of discontinuity:
Removable Discontinuity – If the numerator and denominator can
be factored and common factors divided out of the problem, then
those removed factors are removable discontinuity. On a graph,
the x-value that is the solution of the factor is a tiny gap in the
graph. In a standard graphing window, you cannot see the gap.
(You will need to “zoom in” on your calculator to actually see the
discontinuity.)
(Non-Removable) Jump Discontinuity – The graph has one-sided
limits at that x-value, but they are not equal. There is a significant
“break” in the graph at that x-value. The graph cannot be drawn
without picking up a pencil.
(Non-Removable) Infinite Discontinuity – The graph has one-sided
limits that are ± ∞ . There is an asymptote at that x-value.
Oscillating Discontinuity – The function oscillates rapidly between
1
two fixed values. Example: y = sin
x
Intermediate Value Theorem (p.83)
If f (x ) is continuous on [a, b] , then f (x ) contains all x -values between a and b , and all y -values between
f (a) and f (b) .
AP Calculus AB/BC
Unit 1: Limits and Continuity
Unit One References
Essential Skill
1.1 The student will be able to describe the concept of a
limit (both one-sided and two-sided) in his/her own
words, and explain how a limit can fail to exist.
1.2.a The student will be able to evaluate a limit
graphically.
1.2.b The student will be able to evaluate a limit
numerically (from a table of values).
1.2.c The student will be able to evaluate a limit
algebraically (including direct substitution and
indeterminate form).
1.3 The student will be able to evaluate limits involving
infinity.
1.4.a The student will be able to define horizontal
asymptotes and end-behavior using limits.
1.4.b The student will be able to define vertical
asymptotes using limits.
1.5.a The student will be able to define continuity at a
point.
1.5.b The student will be able to describe common
discontinuities in a function.
1.6 The student will be able to explain and apply the
Intermediate Value Theorem.
Text Examples
Sec 2.1
Sec 2.1 p.66 #37-44
Sec 2.1
Sec 2.1 p.66 #7-14, 19-28, 51-54 (parts b & c)
Sec 2.2 p.76 #1-8, 21-26
Sec 2.2 p.76 #35-48
Sec 2.2 p.76 #13-16, 27-30
Sec 2.3 p.84 #11-18, 41-44, 47-50
Sec 2.3 p.84 #19-24
Sec 2.3