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Limit Involving Infinity [  , not the car]
This is sometimes called “end behavior” of a function
Definition of a horizontal asymptote: The line y  L is a horizontal asymptote of the graph of f if
lim f ( x)  L or lim f ( x)  L
x 
x  
 is not a number so do not say that

1

Let’s explore your favorite topic – limits!
Let’s use our TI to explore.
2 x 2  200 x  100
=
x 
x2 1
Example1: lim
Example 2: lim
x 
Example 3: lim
x 
1

x
4x  1
x2  2

2
x2
x  x
Example 4: lim
End behavior
The “easy-squeezy way” to find lim f ( x)
x  
This works well when we have a
polynomial
type of function.
polynomial
To use end behavior method, re-write the rational expression using the term with the greatest degree from
both the numerator and denominator. Cancel if you can because we are considering the limit as x becomes
very large and do not have to worry about dividing by zero.
Example 5:
2 x 2  200 x  100
lim
x 
x2 1
Example 6:
lim
x  
4x 1
x2 1
Let’s try some more:
5 x 2  3x  7
x 
x7
lim
Example 7:
If ∞ is not given as a multiple-choice answer, then choose the “does not exist” option.
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Example 8:
13  11x 2
x  x 2  26
Example 9:
lim
lim
3x  x 5 / 2
x  7  x 3 / 2
Tricky Ones:
lim cos x 
x 
Because the graph of f(x) oscillates between y = -1 and y = 1, the limit does not exist.
Likewise, lim sin x 
x 
does not exist
And the other trig functions will follow [they can all be re-written in terms of sinx and cosx
cos x
x 
x
Now consider lim
The numerator oscillates between y = -1 and y = 1. BUT the denominator grows larger and larger.
f ( x)
Hence,
cos x
.
x
Notice that the function equals zero and infinite number of times, but we still
have a horizontal asymptote.
4
Let’s try some:
3
3x 2
x2  2
2x
4.
f ( x) 
5.
x
f ( x)  2
x 2
6.
x2
f ( x)  2  4
x 1
7.
f ( x) 
4 sin x
x2 1
8.
f ( x) 
.
f ( x) 
All of the rules of limits still apply!
lim 4 
x 
3

x
x 4 
lim   2 
2 x 
x  
x2  2
2 x 2  3x  5
x2 1