1.5 infinite limits and 3.5 limits at infinity ap calculus i ms. hernandez (print in grayscale or...
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1.5 Infinite Limitsand 3.5 Limits at Infinity
AP Calculus I
Ms. Hernandez
(print in grayscale or black/white)
AP Prep Questions / Warm Up
No Calculator!
(a) 1 (b) 0 (c) e (d) –e (e) Nonexistent
(a) –1/4 (b) –1/2 (c) 0 (d) 1 (e) DNE
1
lnlimx
x
x
22
( 2)lim
4x
x
x
AP Prep Questions / Warm Up
No Calculator!
(a) 1 (b) 0 (c) e (d) –e (e) Nonexistent
(a) –1/4 (b) –1/2 (c) 0 (d) 1 (e) DNE
1
ln ln1 0lim 0
1 1x
x
x
22 2 2
( 2) ( 2) 1 1lim lim lim
4 ( 2)( 2) ( 2) 4x x x
x x
x x x x
1.5 Infinite LimitsVertical asymptotes at x=c will give you
infinite limitsTake the limit at x=c and the behavior of
the graph at x=c is a vertical asymptote then the limit is infinity
Really the limit does not exist, and that it fails to exist is b/c of the unbounded behavior (and we call it infinity)
Determining Infinite Limits from a Graph
Example 1 pg 81Can you get different infinite limits from
the left or right of a graph?How do you find the vertical asymptote?
Finding Vertical AsymptotesEx 2 pg 82Denominator = 0 at x = c AND the
numerator is NOT zero Thus, we have vertical asymptote at x = c
What happens when both num and den are BOTH Zero?!?!
A Rational Function with Common Factors When both num and den are both zero then
we get an indeterminate form and we have to do something else …
Ex 3 pg 83
Direct sub yields 0/0 or indeterminate form We simplify to find vertical asymptotes but how do
we solve the limit? When we simplify we still have indeterminate form.
2
22
2 8lim
4x
x x
x
2
4lim , 2
2x
xx
x
A Rational Function with Common Factors
Ex 3 pg 83: Direct sub yields 0/0 or indeterminate form. When we simplify we still have indeterminate form and we learn that there is a vertical asymptote at x = -2.
Take lim as x-2 from left and right2
22
2 8lim
4x
x x
x
2
22
2 8lim
4x
x x
x
A Rational Function with Common Factors Ex 3 pg 83: Direct sub yields 0/0 or indeterminate form.
When we simplify we still have indeterminate form and we learn that there is a vertical asymptote at x = -2.
Take lim as x-2 from left and right
Take values close to –2 from the right and values close to –2 from the left … Table and you will see values go to positive or negative infinity
2
22
2 8lim
4x
x x
x
2
22
2 8lim
4x
x x
x
Determining Infinite LimitsEx 4 pg 83Denominator = 0 when x = 1 AND the
numerator is NOT zero Thus, we have vertical asymptote at x=1
But is the limit +infinity or –infinity?Let x = small values close to cUse your calculator to make sure – but
they are not always your best friend!
Properties of Infinite LimitsPage 84
Sum/differenceProduct L>0, L<0Quotient (#/infinity = 0)Same properties for Ex 5 pg 84
lim ( )x cf x
lim ( )x cg x L
lim ( )x cf x
Asymptotes & Limits at InfinityFor the function , find(a)
(b)
(c)
(d)
(e) All horizontal asymptotes(f) All vertical asymptotes
2 1( )
xf x
x
lim ( )x
f x
lim ( )x
f x
0lim ( )x
f x
0lim ( )x
f x
Asymptotes & Limits at Infinity
For x>0, |x|=x (or my x-values are positive)
1/big = little and 1/little = bigsign of denominator leads answerFor x<0 |x|=-x (or my x-values are negative)
2 and –2 are HORIZONTAL Asymptotes
2 1( )
xf x
x
2 1 2 1 1lim ( ) lim lim lim 2 2x x x x
x xf x
x x x
2 1 2 1 1lim ( ) lim lim lim 2 2x x x x
x xf x
x x x
Asymptotes & Limits at Infinity2 1
( )x
f xx
0 0 0 0
2 1 2 1 1lim ( ) lim lim lim 2x x x x
x xf x
x x x
2 1 2 1 1lim ( ) lim lim lim 2 2x x x x
x xf x
x x x
1 12 2 2 limDNEx little
1 12 2 2 limDNEx little
3.5 Limit at InfinityHorizontal asymptotes!Lim as xinfinity of f(x) = horizontal
asymptote#/infinity = 0 Infinity/infinity
Divide the numerator & denominator by a denominator degree of x
Some examplesEx 2-3 on pages #194-195
What’s the graph look like on Ex 3.c Called oblique asymptotes (not in cal 1)
KNOW Guidelines on page 195
2 horizontal asymptotesEx 4 pg 196 Is the method for solving lim of f(x) with
2 horizontal asymptotes any different than if the f(x) only had 1 horizontal asymptotes?
Trig f(x)Ex 5 pg 197What is the difference in the behaviors
of the two trig f(x) in this example?Oscillating toward no value vs
oscillating toward a value
Word Problems !!!!!Taking information from a word problem
and apply properties of limits at infinity to solve
Ex 6 pg 197
A word on infinite limits at infinity
Take a lim of f(x) infinity and sometimes the answer is infinity
Ex 7 on page 198 Uses property of f(x)
Ex 8 on page 198 Uses LONG division of polynomials-Yuck!