This document discusses functions with horizontal asymptotes. It provides examples of rational functions that have the same horizontal asymptote as x approaches positive and negative infinity. It also discusses how to evaluate limits of non-rational functions with horizontal asymptotes by dividing the numerator and denominator by the highest power term of x. Finally, it gives examples of finding the horizontal asymptotes of three different functions.

1. Functions with Two Horizontal Asymptotes
Rational Functions always have the same limit as x approaches infinity
as when x approaches negative infinity. 8
3
7
2.5
6
2
5
1.5
4
1
3
0.5
2
1
-4 -3 -2 -1 1 2 3 4
-0.5
-12 -10 -8 -6 -4 -2 2 4 6 8 10 12
-1
-1
1 2x2  5
-2
f ( x)  2
-1.5
f ( x)  2
-3
x 1
-2
3x  1
-4
-2.5
-5
-3
-6
-7
-8
…but what if the function isn’t rational?

2. 3x  2 3x  2
Ex : lim vs. lim
5
2x 1 2x 1
x   x 
2 2
4
3
2
1
-15 -1 0 -5 5 10 15
-1
-2
-3
-4
-5
To evaluate these limits, the technique is the same as before:
we divide the numerator and denominator by x.

3. 3x  2
lim
2x 1
x  2

4. 3x  2
lim
2x2 1
x  

5. Find any Horizontal Asymptotes:
x 2x
1) f ( x)  2) g ( x) 
x 4
2
9x2  5
 4x  2
3) h( x) 
x 2  3x