Calculus: Limits and Indeterminates
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Questions and Answers

What is the limit of a constant function as x approaches any value?

  • It does not exist.
  • It equals the constant itself. (correct)
  • It equals zero.
  • It approaches infinity.
  • Which property of limits is illustrated by the equation lim (kf(x)) = k lim f(x)?

  • Limit of a sum
  • Limit of a quotient
  • Limit of a difference
  • Constant multiple rule (correct)
  • When determining if a limit exists using one-sided limits, what condition must be satisfied?

  • lim- f(x) must equal lim+ f(x). (correct)
  • lim+ f(x) must be greater than zero.
  • lim- f(x) must be less than lim+ f(x).
  • lim- f(x) must equal zero.
  • What strategy is typically used to resolve indeterminate forms like 0/0?

    <p>Using factorization or multiplying by conjugates. (B)</p> Signup and view all the answers

    In the context of limits at infinity, what happens to the limit of a function if it approaches a negative infinity?

    <p>The limit equals negative infinity. (A)</p> Signup and view all the answers

    How should one choose the function to evaluate when calculating a left-hand limit for a piecewise function?

    <p>Use the function that is less than the limit value. (B)</p> Signup and view all the answers

    What is the result when evaluating lim (f(x)/g(x)) if lim g(x) is not equal to zero?

    <p>The limit is equal to f(a)/g(a). (D)</p> Signup and view all the answers

    For the property lim (f(x) ± g(x)) = lim f(x) ± lim g(x), which of the following is true?

    <p>It applies to all types of limits. (D)</p> Signup and view all the answers

    Study Notes

    Limits

    • When f(x) = k where k is a constant, then lim 𝑓(𝑥) = lim 𝑘 = 𝑘 𝑥→𝑎 𝑥→𝑎
    • When 𝑓(𝑥) = 𝑥 𝑛 , then lim 𝑓(𝑥) = lim 𝑥 𝑛 = 𝑎𝑛 𝑥→𝑎 𝑥→𝑎
    • lim 𝑘𝑓(𝑥) = 𝑘 lim 𝑓(𝑥) 𝑥→𝑎 𝑥→𝑎
    • lim 𝑓 𝑥 ± 𝑔 𝑥 = lim 𝑓(𝑥) ± lim 𝑔(𝑥) 𝑥→𝑎 𝑥→𝑎 𝑥→𝑎
    • lim 𝑓 𝑥 𝑔 𝑥 = lim 𝑓(𝑥) lim 𝑔(𝑥) 𝑥→𝑎 𝑥→𝑎 𝑥→𝑎
    • lim 𝑓 𝑥 lim 𝑓 𝑥 𝑥→𝑎 lim = , lim 𝑔(𝑥) ≠ 0 𝑥→𝑎 𝑔 𝑥 lim 𝑔(𝑥) 𝑥→𝑎 𝑥→𝑎

    Indeterminates and Limits

    • 0/0 is an indeterminate form, it can be solved using factorization, or multiplying with conjugates, especially when dealing with surds.

    One-Sided Limits

    • When x approaches a from the left, it is denoted as x → a− (e.g. 0− = ‒0.000000…01)
    • When x approaches a from the right, it is denoted as x → a+ (e.g. 0+ = 0.000000…01)

    Determining Existence of a Limit

    • For a limit to exist, the left-hand limit must equal the right-hand limit. lim− 𝑓(𝑥) = lim+ 𝑓(𝑥) = 𝐿 𝑥→𝑎 𝑥→𝑎
    • This would then imply that lim 𝑓(𝑥) = 𝐿 𝑥→𝑎
    • For piecewise functions, when finding the limit as x approaches a from the left, use the function defined for values less than a. When finding the limit as x approaches a from the right, use the function defined for values greater than a.

    Infinite Limits

    • When y approaches positive infinity from the right(0+), it is denoted as +∞.
    • When y approaches negative infinity from the right(0+), it is denoted as -∞.
    • When y approaches positive infinity from the left(0-), it is denoted as -∞.
    • When y approaches negative infinity from the left(0-), it is denoted as +∞.

    Limits at Infinity

    • When y approaches positive or negative infinity, then the limit is equal to 0.
    • lim 𝑦 = 0 ±∞

    Infinite Indeterminates and Limits

    • ∞ / ∞ is an indeterminate form and can be solved by dividing every term in the numerator and denominator by the highest power of x in the denominator. lim = lim
      𝑥→−∞ 𝑥2 + 1 𝑥→−∞ 𝑥2 1

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    Chapter 8.1 Limits PDF

    Description

    This quiz covers key concepts related to limits in calculus, including the evaluation of limits for constant and polynomial functions, one-sided limits, and indeterminate forms. Test your understanding of these fundamental principles and see how well you can apply them in different scenarios.

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