AP Calculus BC
Chapter 2 – Student Notes
Common Types of Behavior Associated with the Nonexistence of a Limit
1.
f ( x) approaches a different number from the right side of c than it approaches
from the left side.
2. f ( x) increases or decreases without bound as x approaches c.
3.
f ( x) oscillates between two fixed values as x approaches c.
Definition of Limit
Let f be a function defined on an open interval containing c (except possibly at c) and
let L be a real number. The statement
lim
f ( x)  L
xc
means that for each
  0 there exists a   0 such that if
0 | x  c |  , then | f ( x)  L | 
The Limit of a Function Involving a Radical
Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid
for c>0 if n is even.
n x nc
lim
xc
The Limit of a Composite Function
If
f
and
g
are functions such that
lim
g ( x)  L and lim f ( x)  f ( L),
xc
x L
then
lim
f ( g ( x))  f (lim
g ( x))  f (L)
xc
xc
\
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AP Calculus BC
Chapter 2 – Student Notes
Limits of Trigonometric Functions
Let c be a real number in the domain of the given trigonometric function.
1.
2.
3.
4.
5.
6.
limsin
x  sin c
xc
limcos
x  cos c
xc
lim
tan x  tan c
xc
limcot
x  cot c
xc
limsec
x  sec c
xc
limcsc
x  csc c
xc
The Sandwich or Squeeze Theorem
If h( x)  f ( x)  g ( x) for all x in an open interval containing c, except possibly at c
itself, and if
lim
h( x)  L  lim
g ( x)
xc
xc
then
lim
f ( x) exists and is equal to L.
xc
Two Special Trigonometric Limits
1.
lim sin x  1
x0 x
2.
lim 1 cos x  0
x0
x
Definition of a Horizontal Asymptote
The line
yL
is a horizontal asymptote of the graph of
f
if
lim f ( x)  L or xlim
f ( x)  L

x
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AP Calculus BC
Chapter 2 – Student Notes
Limits at Infinity
If r is a positive rational number and c is any real number, then
lim
x 
c
0
xr
Furthermore, if
xr
is defined when
x  0 , then
lim cr  0.
x
x
Guidelines for Finding Limits at Infinity of Rational Functions
1. If the degree of the numerator is less than the degree of the denominator, then the
limit of the rational function is 0.
2. If the degree of the numerator is equal to the degree of the denominator, then the
limit of the rational function is the ratio of the leading coefficients.
3. If the degree of the numerator is greater than the degree of the denominator, then
the limit of the rational function does not exist.
Definition of Continuity
Continuity at a Point: A function
are met.
f
is continuous at c if the following three conditions
f (c) is defined.
lim
f ( x) exists.
xc
lim
f ( x)  f (c)
xc
1.
2.
3.
Continuity on an Open Interval: A function is continuous on an open interval (a, b) if it
is continuous at each point in the interval. A function that is continuous on the entire real
line ,  is everywhere continuous.


The Existence of a Limit
Let f be a function and let c and L be real numbers. The limit of
approaches c is L if and only if
f ( x) as x
lim f ( x)  L and lim f ( x)  L.
xc
xc
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AP Calculus BC
Chapter 2 – Student Notes
Definition of Continuity on a Closed Interval
A function f is continuous on the closed interval [a, b] if it is continuous on the open
interval (a, b) and
lim f ( x)  f (a) and lim f ( x)  f (b).
xa
The function
xb
f
is continuous from the right at a and continuous from the left at b.
Properties of Continuity
If b is a real number and f and g are continuous at
functions are also continuous at c.
1. Scalar multiple:
x  c , then the following
bf
2. Sum and difference:
f g
fg
f
, if g (c)  0
Quotient:
g
3. Product:
4.
Continuity of a Composite Function
If
by
g
is continuous at c and
f
is continuous at
 f g   x   f  g  x  is continuous at c.
g(c) , then the composite function given
Intermediate Value Theorem
f is continuous on the closed interval [a, b] and k is any number between
f (a) and f (b) , then there is at least one number c in [a, b] such that f (c)  k.
If
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AP Calculus BC
Chapter 2 – Student Notes
Definition of a Vertical Asymptote
f ( x) approaches infinity (or negative infinity) as x approaches c from the right or the
left, then the line x=c is a vertical asymptote of the graph of f .
If
Vertical Asymptotes
f (c)  0, g (c)  0 ,
and there exists an open interval containing c such that g ( x)  0 for all x  c in the
f ( x) has a vertical asymptote
interval, then the graph of the function given by h( x) 
g ( x)
Let
f
and
g
be continuous on an open interval containing c. If
at x=c.
Properties of Infinite Limits
Let c and L be real numbers and let
f
and
g
be functions such that
lim
f ( x)   and lim
g ( x)  L
xc
xc
1. Sum or Difference:
2. Product:
3. Quotient:
 f ( x)  g ( x)   
lim
xc 

 f ( x) g ( x)   , L  0
lim
xc 

 f ( x) g ( x)   , L  0
lim
xc 

lim
xc
g ( x)
0
f ( x)
Similar properties hold for one-sided limits and for functions for which the limit of
f ( x) as x approaches c is  .
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