Limits: One-Sided and Two-Sided; Epsilon-Delta and Error Bars

One-Sided vs Two-Sided Limits

  • The lecture discusses proving a statement about limits and how one-sided limits relate to the two-sided limit.
  • Key idea: to understand the two-sided limit, analyze the left-hand limit (as x approaches c from the left) and the right-hand limit (as x approaches c from the right).
  • A notation cue from the transcript: approaching c means considering values of x such that the distance to c becomes arbitrarily small but x is not exactly equal to c.
    • This is captured by the condition 0 < |x - c| < \delta for some small delta.

Definitions (epsilon-delta style)

  • Two-sided limit:

    • A function $f$ has a limit $L$ as $x \to c$ if for every (\epsilon > 0) there exists a (\delta > 0) such that
      0 < |x - c| < \delta \quad \Rightarrow\quad |f(x) - L| < \epsilon.
    • In words: we can make $f(x)$ fall within any small band around $L$ by taking x sufficiently close to c (but not equal to c).

  • One-sided limits:

    • Right-hand limit: lim⁡x→c+f(x)=L\lim_{x \to c^+} f(x) = L means: for every (\epsilon > 0) there exists (\delta > 0) such that
      0 < x - c < \delta \quad \Rightarrow\quad |f(x) - L| < \epsilon.
    • Left-hand limit: lim⁡x→c−f(x)=L\lim_{x \to c^-} f(x) = L means: for every (\epsilon > 0) there exists (\delta > 0) such that
      0 < c - x < \delta \quad \Rightarrow\quad |f(x) - L| < \epsilon.

Relationship between one-sided and two-sided limits

  • If both one-sided limits exist and are equal to the same value $L$:
    • Then the two-sided limit exists and lim⁡x→cf(x)=L.\lim_{x \to c} f(x) = L.
  • If the left-hand and right-hand limits exist but are different, the two-sided limit does not exist.
  • If either one-sided limit does not exist (finite or infinite), the two-sided limit cannot exist as a finite number (though special cases with infinite limits exist in extended real number sense).

Error bars and the epsilon-delta interpretation

  • The term "error bar" in this context corresponds to bounding the deviation |f(x) - L| by a small (\epsilon).
  • For a target limit $L$ and any (\epsilon > 0), we can find a (\delta > 0) such that when the input x is within (\delta) of c (excluding x = c), the output f(x) lies within (\epsilon) of L:
    0 < |x - c| < \delta \quad \Rightarrow\quad |f(x) - L| < \epsilon.
  • This (\epsilon)-neighborhood around L is the mathematical version of an error bar for the limit.

Examples

  • Example 1: Smooth function with same left and right limits
    • Let $f(x) = x^2$, $c = 3$. Then
      lim⁡<em>x→3−f(x)=lim⁡</em>x→3+f(x)=9,\lim<em>{x \to 3^-} f(x) = \lim</em>{x \to 3^+} f(x) = 9,
      so lim⁡x→3f(x)=9.\lim_{x \to 3} f(x) = 9.
    • For any (\epsilon > 0), choose (\delta > 0) such that if (0 < |x - 3| < \delta) then (|x^2 - 9| < \epsilon).
  • Example 2: One-sided exists, other side does not (no finite two-sided limit)
    • Let $f(x) = \frac{1}{x}$ as $x \to 0$. Then
      lim⁡<em>x→0+f(x)=+∞andlim⁡</em>x→0−f(x)=−∞,\lim<em>{x \to 0^+} f(x) = +\infty \quad \text{and} \quad \lim</em>{x \to 0^-} f(x) = -\infty,
      but the two-sided limit lim⁡x→0f(x)\lim_{x \to 0} f(x) does not exist (no finite value).
  • Example 3: Piecewise function with equal one-sided limits
    • Define $f(x) = \begin{cases} x, & x < 0 \ -x, & x \ge 0 \end{cases}$ and let $c = 0$. Then
      lim⁡<em>x→0−f(x)=0=lim⁡</em>x→0+f(x),\lim<em>{x \to 0^-} f(x) = 0 = \lim</em>{x \to 0^+} f(x),
      hence lim⁡x→0f(x)=0.\lim_{x \to 0} f(x) = 0.
  • Example 4: Discontinuity where the limit exists but the function value does not
    • Let $f(x) = 0$ for $x \neq 0$ and $f(0) = 1$. Then lim⁡x→0f(x)=0\lim_{x \to 0} f(x) = 0 even though $f(0) \neq 0$ (not continuous at 0).

Significance and connections

  • Causality to continuity: If lim⁡x→cf(x)=f(c),\lim_{x \to c} f(x) = f(c), the function is continuous at c.
  • Foundation for derivatives: The derivative is defined via a limit of a difference quotient; both left and right behavior are relevant to the existence of the derivative.
  • Foundational principles: epsilon-delta definitions formalize the intuitive idea of approaching a number from near c on both sides or from a single side.
  • Real-world relevance: Limits underpin numerical approximation, error analysis, and stability of algorithms where approaching a target value from inputs near a point is analyzed.

Practical implications and notes

  • The phrase from the transcript about reaching a limit by considering 0 < |x - c| < δ underpins both one-sided and two-sided definitions.
  • The concept of an "error bar" in this context is the epsilon bound around the limit value L, which can be achieved by choosing an appropriate delta.
  • Terminology to remember:
    • lim_{x \to c^-} f(x) = L (left-hand limit)
    • lim_{x \to c^+} f(x) = L (right-hand limit)
    • lim_{x \to c} f(x) = L (two-sided limit)

Quick recap

  • One-sided limits describe approaching c from a single side.
  • Two-sided limit exists iff both one-sided limits exist and are equal.
  • The epsilon-delta framework provides a precise way to bound the deviation |f(x) - L| by choosing x sufficiently close to c, yielding an interpretive "error bar" for the limit.
  • These ideas are foundational for continuity, derivatives, and many real-world applications of calculus.