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2.2

2.2. Limits at Infinity: Horizontal Asymptotes Infinite Limits: Vertical Asymptotes. Limits when x Increases Without Bound. Let f be a function defined on some interval (a, ∞). Then Means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large.

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2.2

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  1. 2.2 Limits at Infinity: Horizontal Asymptotes Infinite Limits: Vertical Asymptotes

  2. Limits when x Increases Without Bound • Let f be a function defined on some interval (a, ∞). Then Means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large.

  3. Formal Definition

  4. Limits when x Decreases Without Bound • Let f be a function defined on some interval (-∞,a). Then Means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large negative.

  5. Formal Definition

  6. Horizontal Asymptote • The line y = b is called a horizontal asymptote of the curve y = f(x) if either or

  7. Example Solution

  8. Crossing Horizontal Asymptotes • Unlike vertical asymptotes that can never be crossed, horizontal asymptotes are about behavior for LARGE x values. • Behavior close to the origin may cross the horizontal asymptote. • What is important is what happens as x gets very large in the positive or negative direction.

  9. Multiple Horizontal Asymptotes • If the limits are different depending on the direction, then there may be more than one asymptote – • One asymptote will occur as x∞ • Another may occur as x - ∞ An example would be:

  10. Limit Laws • The limit laws used for definite numbers can also be used for finding limits as x±∞.

  11. Infinite Limit of 1/x • It is important to note that and

  12. Infinite Limit of when r > 0 • If r > 0 is a rational number, then • If r > 0 is a rational number such that is defined for all x, then

  13. Agenda – Monday, August 13th • Finding limits analytically hw quiz • Finish notes on limits at infinity & horizontal asymptotes • 2.1: 8,10,12,14,16,18,20,22,24,26,28,31,32, 33,34,35,36,37,38,39,40,41,42,43,44,49,50,51,54,58,63 due 08/14

  14. Finding Horizontal Asymptotes of Rational Functions • Divide both numerator and denominator by the variable with the highest exponent in denominator. • NOTE: If you must deal with a radical, remember that when you divide inside the radical you are dividing by the highest exponent times the index • Simplify. • Take the limit of each term in the numerator and denominator. • Simplify.

  15. Example Solution Find the highest degree n of the terms, then multiply the numerator and the denominator by

  16. Simplify Take limit of each term

  17. Example Solution

  18. Example Solution

  19. Example

  20. Beware of Indeterminate Forms • Indeterminate forms involve ∞’s and 0’s from which no conclusion can be drawn. Examples are: • ∞-∞ • ∞/∞ • 0●∞ • ∞^0 • 1^∞ • 0/0 • 0^0

  21. Going Beyond the Indeterminate Form • When encountering an indeterminate form, it is necessary to try “trickery” to find the limit. • Try “tricks” learned earlier when finding limits such as rationalization, reduction, squeeze theorem, etc. • Ask: Is either term increasing more quickly? Then that term will dominate the answer.

  22. Using Asymptotes and Intercepts to Sketch Graphs • Find the vertical asymptote and draw that with a dotted line. Label. • Find the Horizontal asymptote and draw that with a dotted line. Label. • Find the x-intercepts using algebra (factoring, quadratic formula etc.). Plot these. • Remember your rules of multiplicity: • If the intercept factor is raised to a even power it touches x and returns, • If it is raised to an odd power, it passes through.

  23. Vertical Asymptotes: Infinite Limits • When the limit moves unchecked to either + or - ∞, the limit does not exist. • However, we will use + or - ∞ to denote the function is increasing (decreasing) without bound in either the + or - direction.

  24. What happens as x approaches 0? As x approaches 0 from the left, the equation values increase without bound As x approaches 0 from the right, the equation values increase without bound.

  25. Example

  26. Vertical Asymptote • If the function increases without bound toward either + or - ∞ when x approaches a from either the right or left–hand side, the imaginary line at x = a is a vertical asymptote.

  27. Does this function have a vertical asymptote? Yes, at x =0

  28. Does this function have a vertical asymptote? Yes, at x =0

  29. Example Solution So x=2 is definitely a vertical asymptote

  30. Final Thought It is important to remember that infinite limits are not true limits, they only give an indication of a function’s behavior.

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