Chapter 2 Section 4
When using limits
of infinity or limits that exceed all bounds we use two basic functions:
y = x and y = 1/x
then use algebraic combinations.
| lim | 1 | = 0 | and | lim | 1 | = 0 |
| x®¥ | x | x®-¥ | x |
| lim | 1 | = ¥ | and | lim | 1 | = -¥ |
| x®0+ | x | x®0- | x |
Examples:
| 1. lim | (8 - | 1 | ) ® | lim 8 | - | lim | 1 |
| x®¥ | x | x®¥ | x®¥ | x |
|
8 |
- |
0 = |
8 | ||||
| 2. lim | ( | 6 | )® | lim 6 | · | lim | 1 | · | lim | 1 |
| x®-¥ | x2 | x®-¥ | x®-¥ | x | x®-¥ | x |
|
6 |
· |
0 |
· |
0 = |
0 | |||
End Behavior Model
means to see what happens to f(x) as x®¥
and to notice what function f(x) imitates.
In polynomial functions, end behavior model it the leading term.
EX: f(x) = 8x3 -5x2 + 6x -10
end behavior model is 8x3
| lim | 8x3 -5x2 + 6x -10 | |
| x®±¥ | 8x3 |
| lim | ( | 8x3 | - | 5x2 | + | 6x | - | 10 | ) |
| x®-¥ | 8x3 | 8x3 | 8x3 | 8x3 |
| simplified: | ( | 1 | - | 5 | + | 3 | - | 5 | ) |
| 8x | 4x2 | 4x3 |
|
1 |
- |
0 |
+ |
0 |
- |
0 = |
1 | |
Horizontal and Vertical Asymptotes
Where the graph is undefined.
A line y = b is a horizontal asymptote of the graph y = f(x) if:
| lim | f(x) = b | or | lim | f(x) = b |
| x®¥ | x®-¥ |
A line y = a is a vertical asymptote of the graph y = f(x) if:
| lim | f(x) = ±¥ | or | lim | f(x) = ±¥ |
| x®a- | x®a+ |
Finding Asymptotes: we study the degrees of numerator and denominator.
Example:
| f(x) | = | -x |
| 3x+5 |
Horizontal Asymptote - divide all parts by the highest power of x in the denominator.
| lim | -x |
= |
lim | -1 |
= |
y = | -1 | |||
| x | ||||||||||
| x®¥ | 3x | + | 5 | x®¥ | 3+ | 5 | 3 | |||
| x | x | x | ||||||||
Vertical
Asymptote - set denominator = 0. Find where the function is undefined.
3x+5 = 0 x = -5/3
 |