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Calculus: Limits

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Terms in this set (20)

  • What is the definition of a limit in calculus?
    The limit of a function f(x) as x approaches a value c is the value that f(x) gets closer to as x gets closer to c.
  • How is the limit of f(x) as x approaches c written symbolically?
    It is written as \(\lim_{x \to c} f(x)\).
  • What does it mean if \(\lim_{x \to c} f(x) = L\)?
    It means as x approaches c, the values of f(x) approach the number L.
  • What is a one-sided limit?
    A one-sided limit considers the value of f(x) as x approaches c from only one side: left (\(x \to c^-\)) or right (\(x \to c^+\)).
  • When does a limit not exist?
    A limit does not exist if the left and right limits are not equal, or if the function grows without bound near c.
  • What is the limit of a constant function f(x) = k as x approaches any value c?
    The limit is k, since the function value is always k.
  • How do you find the limit of a polynomial function as x approaches c?
    Substitute x = c directly into the polynomial to find the limit.
  • What is the limit of \(\frac{1}{x}\) as x approaches 0 from the right?
    The limit is +∞ because values become very large positive numbers.
  • What is the limit of \(\frac{1}{x}\) as x approaches 0 from the left?
    The limit is -∞ because values become very large negative numbers.
  • What is the Squeeze Theorem?
    If f(x) ≤ g(x) ≤ h(x) near c and \(\lim_{x \to c} f(x) = \lim_{x \to c} h(x) = L\), then \(\lim_{x \to c} g(x) = L\).
  • What is an indeterminate form in limits?
    An expression like \(\frac{0}{0}\) or \(\infty - \infty\) that does not directly determine the limit.
  • How can limits involving indeterminate forms be evaluated?
    By algebraic manipulation, factoring, rationalizing, or applying L'Hôpital's Rule if applicable.
  • What is the limit of \(\sin x / x\) as x approaches 0?
    The limit is 1.
  • What is the limit of \((1 + \frac{1}{n})^n\) as n approaches infinity?
    The limit is the mathematical constant e, approximately 2.718.
  • What does it mean for a function to be continuous at a point c in terms of limits?
    f is continuous at c if \(\lim_{x \to c} f(x) = f(c)\).
  • What is the limit of \(x^2\) as x approaches 3?
    The limit is 9, since \(3^2 = 9\).
  • How do you interpret \(\lim_{x \to \infty} f(x)\)?
    It describes the behavior of f(x) as x grows without bound.
  • What is the limit of \(\frac{1}{x}\) as x approaches infinity?
    The limit is 0.
  • What is the limit of \(\frac{1}{x}\) as x approaches negative infinity?
    The limit is 0.
  • What is the difference between a limit and a function value?
    A limit describes the value f(x) approaches near c, while the function value is f(c) itself.