INFINITE LIMITS.

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Presentation transcript:

INFINITE LIMITS

LIMITS What is Calculus? What are Limits? Evaluating Limits Graphically Numerically Analytically What is Continuity? Infinite Limits This presentation - When the value of the function grows without bounds - When x approaches

Infinite Limits The statement means that the function grows positively without bounds as x approaches c. The statement means that the function grows negatively without bounds as x approaches c. and One sided limits can also be infinite: The equals signs in the statements above do not mean that the limits exist. On the contrary, it says that the limits fail by demonstrating unbounded behavior as x approaches c. Important note:

Infinite Limits & Rational Functions Infinite limits occur at vertical asymptotes. Rational functions that cannot be fully simplified generate vertical asymptotes. We will limit (no pun intended) our study of infinite limits to rational functions.

Infinite Limits: Examples 1. 2. 3. VA @ x = 1: VA @ x = -2: Hole @ x = 2: (Removable point discontinuity) Infinitely many VA @ : For ea VA, x = c, , For each example, Identify any vertical asymptotes (be sure to simplify the function first to discount any point discontinuities). Graph the function and observe the behavior of the function as it approaches these x = c values from both directions. Does it grow without bound positively or negatively?

Evaluating Limits When x Approaches is asking about the right hand behavior of the function is asking about the left hand behavior of the function Examples: 1. 2. This is an odd polynomial with a positive leading coefficient. So the RH behavior And the LH behavior This is a rational function with a horizontal asymptote at y = 2/3. Therefore: Therefore:

Infinite Limits: Summary Infinite limits are written as The equals sign is misleading since the limit does not exist. Infinite limits generally occur at the vertical asymptotes of rational functions. The function may grow positively or negatively on either side of the asymptote. When asking for a limit as x , i.e., Check the end behavior of the function.