Download Name: Date: 1.3 Guided Notes ~ Evaluating Limits Analytically

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1.3 Guided Notes ~ Evaluating Limits Analytically
Date: _______________________________
Calculus IB-SL
Objective:
1) Use properties of limits to evaluate them analytically.
A STRATEGY FOR FINDING LIMITS



Attempt to evaluate the limit by direct substitution.
(Remember this will only work if the function is continuous when x = c.)
If the limit of f(x) as x approaches c CANNOT be evaluated by direct substitution try to find a function
that agrees with f for all x values other than x = c. In other words, use the cancellation or
rationalization techniques.
Use a graph or table to reinforce your conclusion
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
lim g ( x)  K
and
x c
x c
 lim b  b

x c
lim x  c
Scalar Multiple
lim [b  f ( x)] = bL
lim x n  c n
xc
Sum or Difference
lim [ f ( x)  g ( x)] = L  K
x c
x c
Power
Product
lim [ f ( x)]n = Ln
lim [ f ( x) g ( x)] = LK
xc

x c
xc
Quotient
L
f ( x)
provided K  0
lim

x c g ( x)
K
 RECALL: Direct substitution applies for polynomial, rational, radical,
composite, and trigonometric functions provided the function is continuous at c.
Example 1
Evaluate each limit if it exists.
A. lim 4
x  7
3x 2  5 x  12
x 1
x 1
D. lim
4
2
B. lim1 x  8x  3x  5
x
x2
4x 2  3
2
E. lim5 tan x
x
C. lim
6
F. lim2 csc x
x
3
THE CANCELLATION TECHNIQUE
 Factor the numerator and denominator
 Divide out common factors
THE RATIONALIZATION TECHNIQUE
Rationalize the numerator of the fraction.
 You should expand the numerator but leave the
denominator in factored form. A term should
divide out.
Example 2
Evaluate each limit.
A. lim5
x
C. lim
x 3
4
2  3x
3
2
x  27 x  8
12 x 2  7 x  10
4x  5
B. lim
3
x 1  2
x3
1 2  x
x 1
x 1
D. lim
THE SQUEEZE THEOREM
If f(x) < g(x) < h(x) for all x in an open interval containing c, except possibly at c
itself, and if:
lim f ( x)  L  lim h( x)
x c
x c
then lim g ( x)  L .
x c
SPECIAL TRIGONOMETRIC LIMITS
sin x
lim
1
x 0
x
1  cos x
0
x 0
x
lim
Example 3
 1
Graph y = x2, y = –x2 and f(x) = x 2 cos 2  in the same viewing screen.
x 
Use the graphs to visually observe the Squeeze Theorem and find lim f ( x ) .


x 0






Example 4
Evaluate the following:
3(1  cos x)
A. lim
x 0
x
cos x tan x
x 0
x
B. lim
PRACTICE PROBLEMS
Evaluate each limit analytically.
sec   1
x   sec 
3. lim
cos x  1
x 0
x2
6. lim
sin 5 x
x0
x
2. lim
x2
x  2 x 2  4
5. lim
1. lim
4. lim
x4
x5 3
x4
x2  4
x 2 x 3  8


7. lim
1  cos r 2
r 0
r
x 3  125
x  5
x5
3x  5
x2 5 x  3
8. lim
9. lim
1
10. lim x sec x
x 
11. lim
x 0
1 x
x
1