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Limits at Infinity Horizontal Asymptote - Sonoma Valley High School

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AP Calculus. Limits at Infinity and Asymptotes. Limits at Infinity. Horizontal Asymptote. Limits at Infinity. 1) Graphically. 2) Numerically. -∞ . . . -100. -10. -1. 0 1. 10.
AP Calculus

Limits at Infinity and Asymptotes

Limits at Infinity

1) Graphically

2) Numerically

Horizontal Asymptote

-∞

...

-100

-10

-1

0

1

10

3



3

Example:

Numerator and Denominator both increase without bound as x increases towards ∞. (Indeterminate Form) CHANGE EQUATION

Divide by largest power of x.

Determining Limits at Infinity for Rational Functions

Example: (2 horizontal asymptotes)

Divide by x2

Divide by x2

Divide by x3

Largest power in denominator

Same power in numerator and denominator

Largest power in numerator

Lim = 0

Lim = ratio of leading coefficients

Lim DNE (unbounded behavior)

Horizontal Asymptote at y=0

Horizontal Asymptote at y = 2/3

No horizontal asymptote

With rational functions the horizontal asymptote is the same to the right and to the left. This is NOT true for other types of functions, for example, RADICAL functions.

Trig Functions: (Squeeze Theorem)

Finding Infinite Limits and Slant Asymptotes