1. Lesson 11.4 Limits at Infinity
Essential Question: How find the limits of functions at infinity?

2. Limits at Infinity
Limits at infinity occur when y approaches a number as x approaches infinity.
Visually, the graph approaches a HORIZONTAL ASYMPTOTE.
This provides the range of the function.

3. Recall…
A VERTICAL ASYMPTOTE occurred when y approached infinity as x approached a number.
The vertical asymptotes were where x could not be a value. These were non-removable discontinuities.

4. Where will a horizontal asymptote occur?
A horizontal asymptote occurs when, as x goes to infinity, the power of the numerator is less than or equal to the power of the denominator.

5. Definition of Horizontal Asymptote

6. Short Cuts for Determining Horizontal Asymptotes
If the degree of the numerator is LESS than the degree of the denominator, the horizontal asymptote is y = 0.
The limit equals 0.
If the degree of the numerator is GREATER than the degree of the denominator, there is NO horizontal asymptote.
The limit does not exist here.

7. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is equal to:
The limit equals the ratio of the leading coefficients of the numerator and the denominator.

12. lim 𝑥→∞ 4− 3 𝑥 2
lim 𝑥→∞ 7− 1 𝑥

13. lim 𝑥→∞ −2𝑥+3 3 𝑥 2 +1
lim 𝑥→∞ − 𝑥 𝑥 2 +2

14. How find the limits of functions at infinity?

15. Ticket Out the Door
Evaluate lim 𝑥→∞ 𝑥 2 −2 6 𝑥 2