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    Lesson 5. Limits at Infinity

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    Lesson 5. Limits at Infinity

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    Lesson 5. Limits at Infinity

    Uploaded by

    Dunker

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    © All Rights Reserved
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    Download as pdf or txt
    of 1

    LESSON 5.

    LIMITS at INFINITY

    LIMITS AT INFINITY
    THEOREM 1:
    Let r be any positive integer. Then,
    1 1
    a. lim =0 b. lim =0
    𝑥→+∞ 𝑥𝑟 𝑥→−∞ 𝑥 𝑟

    NOTE: When evaluating the limit of a quotient at infinity, first divide both numerator and
    denominator by the highest power of the variable appearing in the denominator.
    EXAMPLES:
    Evaluate
    1
    1. lim =0
    𝑥→+∞ 𝑥 8

    12
    2. lim = 12 • 0 = 0
    𝑥→+∞ 𝑥 15

    2𝑥 3 3
    2𝑥+3 + 2+ 2+0 2
    𝑥 𝑥 𝑥
    3. lim = lim 4 𝑥 = lim 4 = 0−1 = −1 = −2
    𝑥→+∞ 4−𝑥 𝑥→+∞ 𝑥−𝑥 𝑥→+∞ 𝑥−1

    𝑥2 2𝑥 1 1 2 1
    𝑥 2 +2𝑥+1 + + + + 0+0+0 0
    𝑥3 𝑥3 𝑥3 𝑥 𝑥2 𝑥3
    4. lim = lim 3 = lim 4 = =1=0
    𝑥→−∞ 𝑥 3 +4 𝑥→−∞ 𝑥 + 4 𝑥→−∞ 1+ 3 1+0
    𝑥3 𝑥3 𝑥

    𝑥
    𝑥 𝑥 1 1 1
    5. lim = lim = lim = =1=1
    𝑥→+∞ √𝑥 2 −1 𝑥→+∞ 𝑥2 1
    √ 2− 2 𝑥→+∞ √1− 1 √1−0
    𝑥 𝑥 𝑥2

    THEOREM 2:
    +∞, 𝑟 𝑖𝑠 𝑒𝑣𝑒𝑛
    a. lim 𝑥 𝑟 = +∞ b. lim 𝑥 𝑟 = {
    𝑥→+∞ 𝑥→−∞ −∞, 𝑟 𝑖𝑠 𝑜𝑑𝑑
    EXAMPLES:
    𝑥4 1 1
    𝑥 4 +1 + 𝑥2 + 2 𝑥 2 +0 𝑥2
    𝑥2 𝑥2
    1. lim = lim 2 3𝑥 = lim 3
    𝑥
    = lim = lim = lim 𝑥 2 = + ∞
    𝑥→−∞ 𝑥 2 +3𝑥 𝑥
    𝑥→−∞ + 𝑥→−∞ 1+ 𝑥→−∞ 1+0 𝑥→−∞ 1 𝑥→−∞
    𝑥2 𝑥2 𝑥

    3 2
    2. lim (−6𝑥 5 + 3𝑥 2 − 2) = lim 𝑥 5 (−6 + − )
    𝑥→+∞ 𝑥→+∞ 𝑥3 𝑥5
    = +∞(−6 + 0 − 0)
    = +∞(−6)
    = −∞

    BASIC CALCULUS

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