Download Calculus Ch1 Review – Limits Behavior Associated with

Calculus Ch1 Review – Limits
Behavior Associated with 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.
Theorem 1.1: Some Basic Limits. Let b and c be real numbers and let n be a positive integer.
2.lim x  c
1.lim b  b
3.lim xn  cn
x c
x c
xc
Theorem 1.2 Properties of Limits. Let b and c be real numbers, let n be a positive integer, and let f and
g be functions with the following limits.
lim f ( x)  L and lim g ( x)  K
x c
x c
1. Scalar multiple:
lim bf ( x)  bL
x c
2. Sum or Difference: lim  f ( x)  g ( x)  L  K
xc
3. Product:
4. Quotient
5. Power:
lim  f ( x) g ( x)  LK
xc
 f ( x)  L
provided K≠0
lim 

x c g ( x ) 

 K
n
lim  f ( x)   Ln
x c
Strategies for Finding Limits as x approaches c
1. Try direct substitution if the function is a polynomial, rational, or trig.
2. If direct substitution causes division by zero, then
a. try factoring and if a common factor cancels the denominator then direct substitution
b. try rationalizing the numerator
3. If these both fail, then look at a graph and a table of values as x approaches c
Theorem 1.9 Two Special Trig Limits
sin x
1  cos x
1.lim
1
2.lim
0
x 0
x 0
x
x
Definition of Continuity at a Point:A function f is continuous at c if the following 3 conditions are met.
1. f(c) is defined. 2. lim f ( x ) exists 3. lim f ( x ) = f(c)
x c
x c
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.
If a function f is defined on an interval I (except possibly at c), and f is not continuous at c, then f is
said to have a discontinuity at c. A discontinuity is said to be removable if f can be made continuous
by appropriately defining (or redefining) f(c). Otherwise, the discontinuity is said to be nonremovable.
Theorem 1.10 The Existence of a Limit: Let f be a function and let c and L be real numbers. The limit
of f(x) as x approaches c is L if and only if lim f ( x)  L and lim f ( x)  L
x c
x c
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 b
x a
Intermediate Value Theorem: If f is continuous on the closed interval [a,b], f(a) ≠ f(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.
Properties of Continuity: If b is a real number and f and g are continuous at x = c, then the following
functions are also continuous at c: 1. bf 2. f±g 3. fg 4. f/g
Definition of a vertical asymptote: If f(x) approaches infinity (or negative infinity) as x approaches c
from the right or the lefet, then the line x = c is a vertical asymptote of the graph of f.
Theorem 1.14 about Vertical Asymptotes: Let f and g be continuous on an open interval containing c.
If f(c) ≠ 0 and g(c) = 0, and there exists an open interval containing c such that g(x) ≠ 0 for all x ≠ c in
f ( x)
the interval, then the graph of the function given by h( x) 
has a vertical asymptote at x = c.
g ( x)
Theorem 1.15 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
1. lim  f ( x)  g ( x)  
x c
xc
2. lim  f ( x) g ( x)   if L > 0
x c
3. lim  f ( x) g ( x)   if L < 0
xc
 g ( x) 
4. lim 
0
x c
 f ( x) 
p. 91 9,10,15,17,27-31, 35,37,38,53