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Calculus: Limits and Continuity

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  • What is a limit of a function?
    A limit is the value a function f(x) approaches as x gets really close to a given value.
  • How can limits be found numerically?
    By creating a table of values for x values approaching the target point from both sides and observing the function values.
  • What is the graphical interpretation of a limit?
    The limit is the value that the function values approach on the graph as x approaches a specific point from both sides.
  • Does the limit always equal the function value at that point?
    No, the limit may not be equal to the function value f(c) at the point c.
  • What is a one-sided limit?
    A one-sided limit is the limit of a function as x approaches a point from only the left or only the right side.
  • How is the left-hand limit denoted?
    The left-hand limit is written as \(\lim_{x \to c^-} f(x)\).
  • How is the right-hand limit denoted?
    The right-hand limit is written as \(\lim_{x \to c^+} f(x)\).
  • When does a limit not exist (DNE)?
    If the left-hand and right-hand limits are not equal, the limit does not exist.
  • What is the direct substitution method for finding limits?
    For many functions like polynomials, the limit as x approaches c is simply f(c), found by substituting c into the function.
  • How to find limits of rational functions when denominator ≠ 0?
    The limit equals the function value at that point, found by direct substitution.
  • How to find limits of rational functions when denominator = 0?
    If numerator and denominator both zero, factor and simplify to cancel common factors, then substitute.
  • What is the conjugate method for limits involving radicals?
    Multiply numerator and denominator by the conjugate to eliminate radicals, then simplify and evaluate the limit.
  • What are discontinuities in a function?
    Points where the function has a jump, hole, or vertical asymptote, causing the function to be not continuous.
  • How to determine continuity algebraically?
    Check if the limit exists at a point and equals the function value at that point.
  • What is the difference between limit and function value at a point?
    The limit describes behavior as x approaches a point, while the function value is the actual value at that point.
  • What is the significance of one-sided limits in continuity?
    A function is continuous at a point only if both one-sided limits exist and are equal to the function value.
  • How to use a graph to estimate limits?
    Observe the y-values the function approaches as x approaches the target from left and right.
  • What is the limit notation for approaching from the left?
    Use \(\lim_{x \to c^-} f(x)\) to denote approaching c from the left.
  • What is the limit notation for approaching from the right?
    Use \(\lim_{x \to c^+} f(x)\) to denote approaching c from the right.
  • What happens if the left and right limits differ?
    The overall limit at that point does not exist (DNE).