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INFINITE LIMITS

INFINITE LIMITS. ↑↓. When the y-values grow without bounds . When the x-values approach ±∞ [end behavior] . ←→. The statement means that the function grows positively without bounds as x approaches c . . The statement means that the function

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INFINITE LIMITS

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  1. INFINITE LIMITS ↑↓ • When the y-values grow without bounds. • When the x-values approach ±∞ [end behavior] ←→

  2. 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. One sided limits can also be infinite: Important note: and 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. Infinite Limits: f(x)→↑↓ as x→c

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

  4. (Removable point discontinuity) Infinitely many VA @ : For ea VA, x = c, , Infinite Limits: Examples VA @ x = 1: VA @ x = −2: Hole @ x = 2: • 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? [You can also inspect or test the table values instead of graphing]

  5. Infinite Limits After explaining to a student through various lessons and examples that ∞ I tried to check if he really understood that, so I gave him a different example. This was the result: 5 This is a math joke…..

  6. 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 negative leading coefficient. So the RH behavior And the LH behavior Evaluating Limits When x approaches This is a rational function with a horizontal asymptote at y = 2/3. Therefore: Therefore:

  7. Infinite Limits: Summary • Infinite limits are written as • The equals sign is misleading since the limit does not exist. • Infinite limits occur at the vertical asymptotes of rational or other types of functions. The function may grow positively or negatively on either side of the asymptote. • These are nonremovable discontinuities. • When asking for a limit as x , i.e., • Check the end behavior of the function to determine the value that the function is approaching. • This could be ±∞ if it is growing without bound or a real number if it is approaching a horizontal asymptote.

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