Limits at Infinity (End Behavior)

1. Limits at Infinity(End Behavior) Section 2.3

2. Some Basic Limits to Know • Let’s look at the graph of • What happens at x approaches -∞? • What happens as x approaches ∞? • On the original graph, where is the horizontal asymptote? • General rule: • If , then there is a horizontal asymptote at y = L. (Think about why this is.)

3. Limits Using Degrees • Let’s look at pg 125. • Pay close attention to the blue boxes! • *Big fact: • Multiplying xn by a negative number will reverse the sign as well as the end behavior! • Ex:

4. Limits Using Degrees • These “tricks” also apply to polynomials! • Just look at the degree to find the limit. • Ex: • Ex:

5. Rational Functions • Long way: Divide each term by the highest power of x in the denominator. • Ex: • Divide everything by x. • Simplify: • Use the limit just learned!

6. Rational Functions • Let’s try this one on your own…. • Ex: • Answer should be … • 0

7. Rational Functions • One more… • Ex: • Oh no! What is the issue? • Recall the 4 steps for limits… • So:

8. Rational Functions • Fast way: Use degrees to find end behavior! • Use the term with the highest power in numerator and denominator to decide behavior. • Ex: • Ex: • Ex:

9. Limits With Radicals • Same idea, just watch the radical signs! • Ex: • Only change here is the answer is under the radical:

10. Limits With Radicals • Gets harder when only one side of the fraction is under a radical sign. • Ex: Still divide everything by x! • Why are we using x2 in the numerator • Let’s finish:

11. Limits With Radicals • New twist… • Ex: What is different? • Now we need to divide by -∞. Why? • Again, why x2 in the radical? • Let’s finish:

12. Limits With Radicals • Let’s really have a challenge! • Ex: Remind you of something? • Let’s make this a rational function: • Let’s finish: