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Limits at Infinity quiz

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  • What does the limit of 1/x approach as x approaches infinity?
    The limit approaches 0 as x approaches infinity.
  • When evaluating the limit as x approaches infinity for 5/x, what value does the function approach?
    The function approaches 0 as x approaches infinity.
  • What happens to the function sin(x) as x approaches negative infinity?
    The function oscillates between 1 and -1, so the limit does not exist.
  • What is the general strategy for solving limits at infinity for rational functions algebraically?
    Divide each term in the numerator and denominator by the highest power of x present in the function.
  • What shortcut can you use to quickly determine the limit at infinity for a rational function?
    Compare the highest powers of x in the numerator and denominator to decide the limit's behavior.
  • If the highest power of x is greater in the denominator than in the numerator, what does the limit approach as x approaches infinity?
    The limit approaches 0.
  • If the highest power of x is greater in the numerator than in the denominator, what happens to the limit as x approaches infinity?
    The limit does not exist because the function approaches positive or negative infinity.
  • When the highest powers of x in the numerator and denominator are equal, how do you find the limit at infinity?
    The limit is the ratio of the leading coefficients of the highest power terms.
  • Does the shortcut for limits at infinity work for both positive and negative infinity?
    Yes, the shortcut works for both positive and negative infinity.
  • Why does a rational function with a higher degree in the denominator approach 0 as x approaches infinity?
    Because the denominator grows much faster than the numerator, making the whole fraction approach 0.
  • What is the limit as x approaches infinity for the function (3x^2 + 4x - 1)/(x^3 + 27)?
    The limit is 0.
  • What is the limit as x approaches infinity for the function (2x^4 - x)/(x^2 + 5)?
    The limit does not exist because the numerator's degree is higher than the denominator's.
  • What is the limit as x approaches negative infinity for the function (2x + 1)/(5x - 1)?
    The limit is 2/5, the ratio of the leading coefficients.
  • What does it mean if a function's limit at infinity does not exist?
    It means the function grows without bound, approaching positive or negative infinity, or oscillates without settling to a value.
  • How do you determine the leading coefficient in a rational function for limits at infinity?
    The leading coefficient is the coefficient of the term with the highest power of x in the numerator or denominator.