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```2.2
Limits Involving Infinity
Graphically

What is happening in the graph below?
Graphically
We can make the following statements:
lim  f ( x )  
x 0
lim  f ( x )  
x 0
ALSO:
lim
x  
 lim f ( x ) DNE
x 0
f ( x)  0
lim f ( x )  0
x 
Vertical Asymptotes

When do vertical asymptotes occur
algebraically?
Denominator = 0
 (a function is undefined…this includes trig
functions)


Using Limits:
A vertical asymptote of x = a exists for a function if
lim  f ( x )  
x a
OR
lim  f ( x )  
x a
Horizontal Asymptotes

A horizontal asymptote of y = b exists if
lim
x  
f ( x)  b
OR lim f ( x )  b
x 
Example:
Identify all horizontal and vertical asymptotes of
f ( x) 
x
x 1
2
Special Limits
Example:
sin x
What is lim
x 
x
If we substitute in ∞, sin ∞ oscillates between -1 and 1, so
we must find another way to show this limit algebraically.
USING SANDWICH THEOREM:
 1  sin x  1
1
x

sin x
x

1
x
Special Limits
0
lim
x 
1
0
sin x
 lim
x 
x
0  lim
x 
x
sin x
x 
 lim
0
x
Therefore, by the Sandwich Theorem,
lim
x 
sin x
x
0
1
x
Special Limits
Example:
sin x
What is lim
x 0
x
lim
x 0
sin x
x
1
Special Limits
Example:
cos x  1
What is lim
x 0
lim
x 0
cos x  1
x
x
0
Limits Involving ±∞

subtracting, multiplying, dividing, constant
multiplying, and using powers for limit also
apply to limits involving infinity. (see pg.
71)
End Behavior

We sometimes want to how the ends of
functions are behaving.
◦ We can use much simpler functions to discuss
end behavior than a complicated one that may
be given.
◦ To look at end behavior, we must use limits
involving infinity.
End Behavior
A function g is an end behavior model for f
if and only if
lim
x  
f ( x)
1
g ( x)
Right-end behavior model when x +∞
Left-end behavior model when x -∞
End Behavior

Show that g(x) = 3x4 is an end behavior
model for
f(x) = 3x4 – 2x3 + 3x2 – 5x + 6.
3x  2 x  3x  5x  6
4
lim
3
x  
2
3x
4
2
1
5
2 

lim  1 
 2 
 4
3
x  
3x x
3x
x 

1
Finding End Behavior Models
Find a right end behavior model for the
function f(x) = x + ex
xe
lim
x 
?
x
1
Notice when x is ∞,
e∞ goes to 0.
If we use a function of g(x) = x in the denominator,
0
we get
lim
x 
xe
x
x
x
x e 
1 

  lim  1 
 lim  

x

x  x
x 
x
xe 



Therefore, g(x) = x is a right hand
behavior model for f(x)
1
Finding End Behavior Models
Find a left end behavior model for the
function f(x) = x + ex
lim
xe
x  
?
x
1
Notice when x is ∞,
ex goes to ∞ and x
goes to –∞.
Which one has more effect on the left-end of
the function? (Which one gets to ∞ faster?)
Therefore, use e–x as a left-end behavior model
for f(x).
e∞
Finding End Behavior Models
Find a left end behavior model for the
function f(x) = x + ex
lim
x  
lim
x  
lim
x  
xe
x
e
1
?
x
e
x
xe
x
xe
x
1
0
x
 x
e
 lim   x   x
x   e
e

1
1




Therefore, e–x is a left-end behavior
model for f(x).
HW
Section 2.2 (#1-7 odd, , 21, 23, 25, 27-33
odd, 39, 41, 43, 45-49 odd)
 Web Assign due Monday night

```
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