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INSTRUCTORS Raffi Hovasapian John Zhu
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For more information, please see full course syllabus of Calculus AB
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Lecture Comments (7)

0 answers

Post by Si Jia Wen on August 19, 2013

Yes, he made a few mistakes in this video. These rules only apply if x is approaching infinity.

0 answers

Post by Linsey Smith on May 7, 2013

I don't think your third example is correct. As the problem approaches zero, don't you put the use the lowest degree to find the limit? You used the highest degree, which would be if x was approaching infinity.

4 answers

Last reply by: Darley Emenim
Mon Jun 30, 2014 4:35 PM

Post by Ivan Murray on June 5, 2012

On the Applying the 3rd Rule problem, should the problem be as the Limit approaches Infinity or as the Limit approaches 0?

Rational Limit Rules

  • Highest x degree in numerator:
  • Highest x degree in denominator:
  • Highest x degree in both numerator and denominator:
  • Straight forward, no need to overcomplicate!

Rational Limit Rules

Find limx → ∞ [(x2 + 5x)/(x + 1)]
  • Higher degree polynomial in the numerator
  • limx → ∞ [(x2 + 5x)/(x + 1)] = ∞
limx → ∞ [(x2 + 5x)/(x + 1)] = ∞
Find limx → ∞ [(x + 1)/(x2 + 5x)]
  • Higher degree polynomial in the denominator
  • limx → ∞ [(x + 1)/(x2 + 5x)] = 0
limx → ∞ [(x + 1)/(x2 + 5x)] = 0
Find limx → ∞ [(5x7 + 200x6 + 999999999)/(2.5x7 − 9999999)]
  • Equal degrees in numerator and denominator. Take ratio of coefficients of highest degrees
  • limx → ∞ [(5x7 + 200x6 + 999999999)/(2.5x7 − 9999999)] = [5/2.5] = 2
  • [5/2.5] = 2
2
Find limx → ∞ [1/(x3 − 3x2)]
  • Highest degree polynomial in the denominator
  • limx → ∞ [1/(x3 − 3x2)] = 0
limx → ∞ [1/(x3 − 3x2)] = 0
Find limx → ∞ [(x2 + 2x11)/(x3 + x11)]
  • Equal degrees in numerator and denominator
  • limx → ∞ [(x2 + 2x11)/(x3 + x11)] = [2/1] = 1
limx → ∞ [(x2 + 2x11)/(x3 + x11)] = [2/1] = 1
Find limx → ∞ [((3x4 + 2x)(5x3 + x2 + 1))/(x7)]
  • limx → ∞ [((3x4 + 2x)(5x3 + x2 + 1))/(x7)] = limx → ∞ [((3x4 + 2x)(5x3 + x2 + 1))/(x7)]
    = limx → ∞ [(15x7 + 10x4 +3x6 + 2x3 + 3x4 + 2x)/(x7)]
  • In truth, we only needed to multiply out the largest exponents, as it's the only one that's going to factor when we're taking this limit to infinity
limx → ∞ [((3x4 + 2x)(5x3 + x2 + 1))/(x7)] = [15/1] = 15
Find limx → ∞ [(√x)/x]
  • limx → ∞ [(√x)/x] = limx → ∞ [1/(√x)] = 0
0
Find limx → ∞ [(x[1/10000])/(1099)]
  • 1099 is a large number. Astronomically large. More than the estimated 1080 atoms in the universe. It still does not matter. Infinity dominates.
  • limx → ∞ [(x[1/10000])/(1099)] = ∞
limx → ∞ [(x[1/10000])/(1099)] = ∞
Find limx → ∞ [(x9 + x5 + 4x)/(−2x9 + 13)]
  • Equal exponents
  • limx → ∞ [(x9 + x5 + 4x)/(−2x9 + 13)] = [1/(−2)] = −[1/2]
limx → ∞ [(x9 + x5 + 4x)/(−2x9 + 13)] = [1/(−2)] = −[1/2]
Find limx → ∞ [(1 − 4x2)/(x − x2)]
  • Equal exponents
  • limx → ∞ [(1 − 4x2)/(x − x2)] = [(−4)/(−1)] = 4
limx → ∞ [(1 − 4x2)/(x − x2)] = [(−4)/(−1)] = 4

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Rational Limit Rules

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Rational Limit Rules 0:07
    • Review of Solving Problem Algebraically
    • Limit Rules
    • Rule 1
    • Rule 2
    • Rule 3
  • Rational Limit Rules 1:02
    • Applying 1st Rule
  • Rational Limit Rules 1:50
    • Applying 2nd Rule
  • Rational Limit Rules 2:26
    • Applying 3rd Rule