Download Properties of Limits

Miss Battaglia
AB/BC Calculus
• Remember that the limit of f(x) as x
approaches c does not depend on the
value of f at x=c… But it might happen!
• Direct substitution
f (x) = f (c) substitute x for c
• lim
x®c
• These functions are continuous at c
Theorem 1.1 Some Basic Limits
Let b and c be real numbers and let n be a positive integer.
lim b = b
1. x®c

2.
lim x = c
x®c
3.
lim x n = c n
x®c
Example: Evaluating Basic Limits
a)
lim 3 =
x®2
b)
lim x =
x®-4
c) lim x =
2
x®2
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
x®c
and
lim g(x) = K
x®c
1. Scalar Multiple:
lim[bf (x)] = bL
2. Sum or difference:
lim[ f (x)± g(x)] = L ± K
3. Product:
lim[ f (x)g(x)] = LK
4. Quotient:
lim
5. Power:
lim[ f (x)]n = Ln
x®c
x®c
x®c
x®c
x®c
f (x) L
= ,
g(x) K
provided K ¹ 0
lim(4x + 3)
2
x®2
Theorem 1.3: Limits of Polynomial and Rational Functions
If p is a polynomial function and c is a real number, then
lim p(x) = p(c)
x®c
If r is a rational function given by r(x)=p(x)/q(x) and c is
a real number such that q(c)≠0, then
p(c)
lim r(x) = r(c) =
x®c
q(c)

Find the limit:
x + x+2
lim
x®1
x +1
2
Theorem 1.4: The Limit of a Rational Function
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.
lim n x = n c
x®c
Theorem 1.5: The Limit of a Composite Function
If f and g are functions such that
and lim f (x) = f (L) , then
x®L
(
lim g(x) = L
x®c
)
lim f (g(x)) = f lim g(x) = f (L)
x®c
x®c
f(x)=x2 + 4
and
f (x)
• Find lim
x®0
g(x)
• Find lim
x®4
g( f (x))
• Find lim
x®0
g(x)=
x
Theorem 1.6: Limits of Trigonometric Functions
Let c be a real number in the domain of the given
trigonometric function.
1.
lim sin x = sinc
2.
lim cos x = cosc
3.
lim tan x = tanc
4.
lim cot x = cot c
lim sec x = secc
6.
x®c
x®c
5.
x®c
Examples:
a.
lim tan x =
b.
lim(x cos x) =
c.
lim sin 2 x =
x®0
x®p
x®0
x®c
x®c
lim csc x = cscc
x®c
Theorem 1.7: Functions that Agree at All But One Point
Let c be a real number and let f(x)=g(x) for all x≠c in an open interval
containing c. If the limit of g(x) as x approaches c exists, then the
limit of f(x) also exists and
lim f (x) = lim g(x)
x®c
x®c
Find the limit: lim x -1
3

x®1
x -1

2
x
+
x
6
Find the limit: lim
x®-3
x +3

x +1 -1
Find the limit: lim
x®0
x
Theorem 1.8: The 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
Theorem 1.9: Two Special Trig Limits
sin x
=1
1. lim
x®0
x
2.
1- cos x
lim
=0
x®0
x
æ1ö
lim x cos ç 2 ÷
x®0
èx ø
2

Find the limit:
tan x
lim
x®0
x

Find the limit:
sin 4x
lim
x®0
x
 Read
1.3
 Page 67 #17-75 every other odd,
85-89, 107, 108, 117-122