Download Limits Involving Infinity

Section 2.2a
Limits Involving Infinity
Note: “Infinity” does not represent a real number……however,
We can say “the limit of f as x approaches infinity,”
meaning the limit of f as x moves increasingly far to the
right on the number line, or…
Saying “the limit of f as x approaches negative infinity”
means the limit of f as x moves increasingly far to the
left.
Limits Involving Infinity
The graph of the reciprocal function:
1
f  x 
x
Our new limits:
1
lim    0
x  x
 
1
lim    0
x  x
 
The line y = 0 is a horizontal
asymptote of the graph of f…
Definition: Horizontal Asymptote
The line y = b is a horizontal asymptote of the graph
of a function y = f(x) if either
lim f  x   b
x 
or
Ex: Find any H.A. of the graph of
lim f  x   b
x
f  x   2  1 x 
lim f  x   lim f  x   2
x 
x 
 H.A.: y = 2
Definition: Horizontal Asymptote
The line y = b is a horizontal asymptote of the graph
of a function y = f(x) if either
lim f  x   b
x 
or
Ex: Find any H.A. of the graph of
Investigate with both a
graph and a table…
lim f  x   b
x
f  x 
x
x 1
lim f  x   1 lim f  x   1
x 
x
 H.A.: y = –1, y = 1
2
Sandwich Theorem Revisited
Find
lim f  x 
x 
for
sin x
f  x 
x
First, what do the graph and table suggest???
Confirm Analytically:
1  sin x  1
And by the Sandwich Theorem:
For x > 0, we have
1 sin x 1
 

x
x
x
sin x
1
 1
0  lim     lim
 lim  0
x 
x  x
 x  x x
Limits Involving Infinity
Properties of Our New Limits:
Note: All of the properties for limits approaching real
numbers also hold for limits approaching infinity!!!
Including  Sum Rule, Difference Rule, Product Rule,
Constant Multiple Rule, Quotient Rule, Power Rule
Limits Involving Infinity
Rewrite:
Find
5 x  sin x
 5 x sin x 
 lim  
lim

x

x 
x 
x
 x
sin x
 lim 5  lim

5

0

5
x 
x 
x
Limits Involving Infinity
Sometimes, a function outgrows all bounds (either
positive or negative) as x approaches a finite number
a  we write:
lim f  x   
x a
or
lim f  x   
xa
Think back to the reciprocal function:
1
lim  
x 0 x
and
1
lim  
x0 x
The line x = 0 is a vertical asymptote of the graph of f…
Definition: Vertical Asymptote
The line x = a is a vertical asymptote of the graph
of a function y = f(x) if either
lim f  x   
xa
or
lim f  x   
xa
Limits Involving Infinity
For the given function, (a) find the vertical asymptotes; (b) describe
the behavior of the function to the left and right of each V.A.
1 x
1 x
f  x  2

2 x  5 x  3  2 x  1 x  3
1
Now, check the graph!
V.A.: x   , x  3
2
lim  f  x   
x 0.5
lim  f  x   
x 0.5
lim f  x   
x 3
lim f  x   
x 3