Limits: Two-Sided Limit and One-Sided Limits (Overview from Transcript)

  • Core idea: a limit describes how the function values behave as x gets arbitrarily close to a specific input c.
  • Setup and terminology:
    • The horizontal axis is x, the input to the function.
    • The vertical axis shows the function values, f(x).
    • The phrase “as x approaches c” means x gets closer and closer to c from either side.
    • The notation for the limit is lim⁡x→cf(x)=L\lim_{x\to c} f(x) = L, which reads: the limit of f(x) as x approaches c equals L.
    • The word form hint: the symbol lim⁡\lim reminds us of the word “approach.”
  • One-sided limits (left and right):
    • Left-hand limit: lim⁡x→c−f(x)\lim\limits_{x\to c^-} f(x)
    • Right-hand limit: lim⁡x→c+f(x)\lim\limits_{x\to c^+} f(x)
    • These express the behavior of f(x) as x approaches c from the left (values less than c) and from the right (values greater than c).
  • Existence condition for the two-sided limit:
    • A two-sided limit lim⁡<em>x→cf(x)=L\lim<em>{x\to c} f(x) = L exists if and only if the left-hand and right-hand limits exist and are equal: lim⁡</em>x→c−f(x)=lim⁡x→c+f(x)=L.\lim</em>{x\to c^-} f(x) = \lim_{x\to c^+} f(x) = L.
    • Intuition: the graph must approach the same y-value from both sides as x gets near c.
  • What that means in words:
    • The limit from the left equals the limit from the right, and that common value is the limit of f as x approaches c.
    • If the left and right limits are not equal, the two-sided limit does not exist.
  • Visual intuition (picture description from the transcript):
    • Draw a point at x = c on the x-axis and a corresponding y-value on the graph, f(c).
    • As x approaches c from the left, the corresponding f(x) values approach some L.
    • As x approaches c from the right, the corresponding f(x) values also approach some (potentially the same) value.
    • If both approaches give the same L, the limit exists and equals L; otherwise, it does not exist.
  • Concrete example from the transcript:
    • Given a graph where at x = 4, f(4) = 6 (the function value at 4 is 6).
    • The left-hand limit as x → 4⁻ is 6:
      lim⁡x→4−f(x)=6.\lim\limits_{x\to 4^-} f(x) = 6.
    • The right-hand limit as x → 4⁺ is also 6:
      lim⁡x→4+f(x)=6.\lim\limits_{x\to 4^+} f(x) = 6.
    • Therefore, the two-sided limit exists and equals 6:
      lim⁡x→4f(x)=6.\lim\limits_{x\to 4} f(x) = 6.
  • Important nuance mentioned (and worth noting):
    • The value of f at c, i.e., f(c), does not determine the limit; the limit depends on the behavior of f(x) as x approaches c.
    • It is possible for f(c) to be different from the limit value or for f(c) to be undefined while the limit exists.
  • Quick summary statements to memorize:
    • If lim⁡<em>x→c−f(x)=lim⁡</em>x→c+f(x)=L\lim\limits<em>{x\to c^-} f(x) = \lim\limits</em>{x\to c^+} f(x) = L, then lim⁡x→cf(x)=L.\lim\limits_{x\to c} f(x) = L.
    • If the left-hand limit and the right-hand limit are not equal, the limit does not exist.
  • Optional practice prompt inspired by the transcript:
    • Given a graph where both sides approach 6 as x → 4, determine the limit:
    • lim⁡<em>x→4−f(x)=6\lim\limits<em>{x\to 4^-} f(x) = 6, lim⁡</em>x→4+f(x)=6\lim\limits</em>{x\to 4^+} f(x) = 6 → lim⁡x→4f(x)=6.\lim\limits_{x\to 4} f(x) = 6.