Limits at Infinity and End Behavior Quiz
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Questions and Answers

Which of the following describes the behavior of a function as x approaches positive infinity?

  • The function approaches negative infinity
  • The function oscillates
  • The function approaches a specific number
  • The function approaches positive infinity (correct)

    • Which of the following describes the behavior of a function as x approaches negative infinity?

  • The function approaches positive infinity (correct)
  • The function oscillates
  • The function approaches negative infinity
  • The function approaches a specific number

    • What is the term used to describe the behavior of a function as x increases without bound?

  • Vertical asymptote
  • End behavior (correct)
  • Oscillation
  • Horizontal asymptote

    • What is the term used to describe the behavior of a function as x decreases without bound?

    <p>End behavior</p> Signup and view all the answers

    If a function approaches a specific number L as x increases without bound, what is the equation of the horizontal asymptote?

    <p>y = L</p> Signup and view all the answers

    Which of the following is a property of exponents?

    <p>Exponents can be any real number</p> Signup and view all the answers

    What is the value of b^0, where b is a nonzero real number?

    <p>1</p> Signup and view all the answers

    Which of the following is a correct expression for b^(p/q), where p/q is a positive rational number expressed in lowest terms?

    <p>b^(1/q) * b^p</p> Signup and view all the answers

    What assumption is made throughout this section regarding the value of b?

    <p>b is positive</p> Signup and view all the answers

    Which of the following is a method for defining irrational powers of b?

    <p>Using successive approximation</p> Signup and view all the answers

    Study Notes

    End Behavior

    • Describes the behavior of a function as x approaches positive or negative infinity
    • As x approaches positive infinity refers to the function's behavior as x becomes increasingly large
    • As x approaches negative infinity refers to the function's behavior as x becomes increasingly small (more negative)
    • End behavior is the term used to describe the behavior of a function as x increases or decreases without bound
    • Horizontal asymptote is the line that the function approaches as x increases or decreases without bound
    • The equation of the horizontal asymptote is y = L if a function approaches a specific number L as x increases or decreases without bound

    Properties of Exponents

    • b^0 = 1 for any non-zero real number b
    • b^(p/q) = (b^(1/q))^p for any positive rational number p/q expressed in lowest terms
    • b^(1/q) = qth root of b for any positive integer q
    • b^(p/q) = qth root of b^p for any positive integers p and q
    • b^n = (b^(1/q))^p for any positive integers p and q
    • b^(p/q) = b^(m/n) = b^r, where r is the simplified rational number form of (mp)/(nq)
    • b^m * b^n = b^(m+n) for any integers m and n
    • (b^m)^n = b^(m*n) for any integers m and n
    • (a*b)^n = a^n * b^n for any non-zero real numbers a and b and any integer n
    • (a/b)^n = a^n / b^n for any non-zero real numbers a and b and any integer n
    • (b^n)^m = b^(n*m) for any non-zero real numbers a and b and any integer n and m
    • The assumption made throughout this section regarding the value of b is that b is a positive real number
    • Irrational powers of b are defined by using limits
    • Irrational powers of a positive real number b are defined by considering the powers of b that are rational numbers
    • The limit of b^r as r approaches the irrational number is defined as the value of b raised to that irrational power
    • The definition of a power with an irrational exponent can be extended to negative values of b, that is, b is a negative number and r is a rational number
    • The value of a power with a negative base and a rational exponent is determined by the sign of the exponent
    • If the exponent is even, the result is always positive
    • If the exponent is odd, the result is always negative
    • The value of a power with a negative base and an irrational exponent is also defined using limits
    • Limits are used to define powers with irrational exponents because they allow us to take the limit of a sequence of rational numbers that approach the irrational number
    • The limit of b^r as r approaches the irrational number is defined as the value of b raised to that irrational power

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    Description

    This quiz tests your understanding of limits at infinity and the end behavior of a function. It covers topics such as the behavior of a function as x approaches positive or negative infinity, and the concept of horizontal asymptotes.

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