Download 2.3 - Calculating Limits Using The Limit Laws

2.3 - Calculating Limits Using
The Limit Laws
1
Basic Limit Laws
1. lim c  c
2. lim x  a
xa
(a, c)


|
a
x a
y=c
(a, a)

y=x
 |
a
3a. lim xn  a n where n is a positive integer.
x a
3b. lim n x  n a where n is a positive integer.
x a
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Limit Laws Generalized
Suppose that c is a constant and the following
limits exist lim f ( x) and lim g ( x).
x a
x a
1. lim  f ( x)  g ( x)  lim f ( x)  lim g ( x)
x a
x a
xa
2. lim  f ( x)  g ( x)  lim f ( x)  lim g ( x)
xa
xa
x a
3. lim cf ( x)  c lim f ( x)
x a
x a
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Limit Laws Generalized
4. lim  f ( x) g ( x)  lim f ( x)  lim g ( x)
xa
xa
x a
f ( x)
 f ( x)  lim
x a
5. lim 

x a g ( x) 
g ( x)

 lim
x a
n
6a. lim  f  x   lim f  x  where n is a positive integer.
x a
 x a

n
6b. lim n f ( x)  n lim f ( x) where n is a positive integer.
x a
x a
4
Examples
Evaluate the following limits. Justify each step
using the laws of limits.


1. lim 3x  2 x  5
2
x  3
 3x  2 
2. lim 

x 1
 x 5 
3. lim x  2 x
3
2
x 2
5
Direct Substitution Property
If f is a polynomial or a rational function and a
is in the domain of f, then
lim f ( x)  f (a)
xa
6
Examples
You may encounter limit problems that seem to be impossible
to compute or they appear to not exist. Here are some tricks to
help you evaluate these limits.
1. If f is a rational function or complex:
a. Simplify the function; eliminate common factors.
b. Find a common denominator.
c. Perform long division.
2. If a root function exists, rationalize the numerator or
denominator.
3. If an absolute values function exists, use one-sided limits
and the definition.
a if a  0
a 
 a if a  0
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Direct Substitution Property
Evaluate the following limits, if they exist.
x3 1
1. lim 2
x 1 x  1
1 
1
3. lim   2 
t 1
t 0  t
1 h 1
2. lim
h
h 0
4. lim
x 2
x2
x2
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Theorem
If f(x)  g(x) when x is near a (except possibly
at a) and the limits of f and g both exist as x
approaches a, then
lim f ( x)  lim g ( x)
xa
xa
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The Squeeze (Sandwich) Theorem
If f(x)  g(x)  h(x) when x is near a (except
possibly at a) and
lim f ( x)  lim h( x)  L
xa
x a
then
lim g ( x)  L
xa
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Example
Prove that lim xesin( / x )  0 is true.
x 0
Strategy To begin,
bind a part of the
function (usually the
trigonometric part if
present) between
two real numbers.
Then create the
original function in
the middle.
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You Try It
Evaluate the following limits, if they exist, in groups of
no more than three members.
10 x  9  1
1. lim
x 1
x 1
2 
 1
2. lim 
 2

x

2
x

2
x
x 2 

t 7
3. lim
t 7 7  t
2x 1
4. lim
x
x 0
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