Download Algebraic Limits Notes

4. Algebraic
Limits
So far …..
• Numerically
• Graphically
• What if you don’t have a graph or a calculator or your
brilliant friend sitting next to you?
• We will look at various algebraic methods. Their names are
not important, but it is important to recognize their forms.
1. Direct Substitution
• This is the method you should ALWAYS try first!
• Example 1
lim
x4
x
x
2
lim csc x
x 5 /3
sin x
lim
x 10
x
2. VA
0
• If you use direct substitution and get
0 , that means
there is a VA and the answer will be DNE or  if it is a
one-sided limit
• Example 2
x3  1
lim
x 1 x  1
lim tan x
x  /2
Indeterminate form
• We say that f(x) is indeterminate at x = c if when we
evaluate f(c), we obtain an undefined expression
0
,
0

,

  0,

• Strategy is to transform f(x) algebraically into a new
expression that is defined and continuous at x=c.
• Remember that if you get something like  0 then this is
NOT indeterminate
0
3. Factor and divide out
• The factor that causes 0/0 in the denominator can be
simplified with its common factor in the numerator and
then direct substitution can be used on the resulting function
• Example 3
x2  4
lim
x2 x  2
x3  1
lim
x 1 x  1
4. Rationalization conjugation
• This method works well where there is a sum or difference
of one or more radical terms. It involves multiplying by
conjugate/conjugate
• Example 4
lim
x4
x 2
x4
lim
x 0
x2  9  3
x2
5. Least common denominator
• This method works well if there is a compound fraction or a
complex fraction (a fraction within a fraction.) It involves
finding the LCD
• Example 5
lim
x 0
x
1
1

6 x6
2 
 1
lim 
 2

x 1
x

1
x

1


6. Expand
• This method works well if there is some obvious math to do
like expanding binomials
• Example 6
( x  h)  x
lim
h 0
h
2
2
( x  h) 2  2( x  h)  1  ( x 2  2 x  1)
lim
h 0
h
7. Trig Manipulation
• We have already seen this method, it involves using trig
identities to rewrite limits to be ones we have memorized
sin nx
1  cos nx
lim
1
lim
0
x 0
x

0
nx
nx
• Example 7
tan x
lim
x  / 2 sec x
lim
x 0
tan 4 x
x
8. General cleverness
• Try this when all else fails.
• Example 8
lim
x 5
3 x  15
x 2  10 x  25
summary
Real #
lim
x
Compare
Exponents
(this is your answer)
#
VA
0 Plug in # to left
lim
lim
x
anything
else
x
anything
else
and/or right
Plug in

0
0
is answer
or other indeterminant
Algebra to simplify
Then plug in