Notes on Limits: Two-Sided, One-Sided, and Function-Value Relationships

Overview: f and g near x = 0

  • Both f and g are undefined at x = 0 because 0 is in their denominator (domain excludes x = 0).
  • Abbreviation for undefined: DNE = does not exist.
  • Intuition: undefined at a point does not stop us from asking about what the function does as we approach that point from either side.
  • Two common viewpoints near x = 0:
    • f(x) near 0 tends to infinity (unbounded growth) → the limit is
      lim⁡x→0f(x)=∞.\lim_{x\to 0} f(x) = \infty.
    • g(x) simplifies to a linear form away from x = 0, so near 0 it behaves like the line y = x + 2, giving a finite limit.

The two key examples (f and g) and their limits

  • Function f:
    • Structure: undefined at x = 0 due to a zero denominator; as x approaches 0 from either side, f(x) grows without bound.
    • Conclusion: the limit does not exist in the finite sense; it diverges to infinity:
      lim⁡x→0f(x)=∞.\lim_{x\to 0} f(x) = \infty.
  • Function g:
    • Algebraic form near x ≠ 0: numerator $x^2 + 2x$, denominator $x$.
    • Simplification for x ≠ 0:
      g(x)=x2+2xx=x+2,x≠0.g(x) = \frac{x^2 + 2x}{x} = x + 2, \quad x \neq 0.
    • Graphically: a straight-line trend with a hole (open circle) at x = 0 and y = 2.
    • Limit from both sides:
      lim⁡x→0g(x)=2.\lim_{x \to 0} g(x) = 2.
    • Note: g(0) is not defined; there is a hole at (0, 2).
  • How this matches the table/table-image idea:
    • Values near zero (e.g., x = 0.1, 0.01, -0.1, -0.01) yield outputs approaching 2, illustrating the limit even though x = 0 is excluded from the domain.

Definition and intuition for limits

  • Informal definition (as stated in class):
    lim⁡x→af(x)=l\lim_{x\to a} f(x) = l
    if, when x is near a (from either side), the values f(x) get arbitrarily close to l.
  • This emphasizes the idea that a limit describes the value the function is approaching, not necessarily the value the function actually takes at a.
  • Two-sided limit (the default):
    • The limit as x approaches a from the left (
      lim⁡<em>x→a−f(x)))mustagreewiththelimitfromtheright(\lim<em>{x\to a^-} f(x))) must agree with the limit from the right (\lim{x\to a^+} f(x))) for the two-sided limit to exist.
  • When a is in the domain, the limit often equals the function value at a (continuity), but this is not required in general.

Relationship between a limit and the function value at the point

  • Example where the limit equals the function value (a is in the domain):
    • A polynomial (e.g., a quadratic defined for all x):
      If f(x)=x2+3x−1, then lim⁡x→1f(x)=f(1)=3.\text{If } f(x) = x^2 + 3x - 1, \text{ then } \lim_{x\to 1} f(x) = f(1) = 3.
  • Example where the limit exists but the function value at a is undefined (a not in the domain):
    • The earlier f-like case: even though f(0) is undefined, the limit as x → 0 might exist or diverge (here it diverges to ∞).
  • Example where the limit exists (finite) but the function value at a differs from that limit (a is a point of discontinuity):
    • A piecewise or redefined function around a point can have a limit that is not equal to the function value at that point.

A concrete piecewise example to illustrate limits vs function values

  • Define a function that is x^2 everywhere except at x = 2, where it’s redefined to be 5:
    • For x ≠ 2: f(x) = x^2
    • For x = 2: f(2) = 5
  • What happens near x = 2?
    • As x → 2 (x ≠ 2), f(x) → 4 (since x^2 → 4).
    • Therefore, the limit is
      lim⁡x→2f(x)=4.\lim_{x\to 2} f(x) = 4.
    • But f(2) = 5, so the value of the function at 2 is different from the limit.
  • Key takeaway: A limit can be about the values near a, independently of the actual value at that point.

A related set of examples: same limit, different function values at a

  • Consider three functions around x = 5:
    • f(5) = 3 (the function value at 5 is 3).
    • g(5) is undefined (open circle at (5, 3)); however, ( \lim_{x\to 5} g(x) = 3 ).
    • h(5) = 1 (closed point at (5, 1)); yet, ( \lim_{x\to 5} h(x) = 3 ).
  • Conclusion: All three can share the same limit as x approaches 5 (namely 3) while having different values or definitions at x = 5.
  • Takeaway: The limit is about behavior near the point, not necessarily the exact function value at that point.

Using tables and graphs to motivate a limit (sine example)

  • Classic limit:lim⁡x→0sin⁡xx=1.\lim_{x\to 0} \frac{\sin x}{x} = 1.
  • Why a table helps: plugging x = 0 gives 0/0 (an undefined form). A table of nearby x-values shows the outputs trending toward 1 from both sides.
  • Calculator tip mentioned in class: ensure the calculator is in gradient/radian mode to get accurate results for small x.
  • Example table entries (near x = 0):
    • x = 0.1 → sin(0.1)/0.1 ≈ 0.9983
    • x = 0.01 → sin(0.01)/0.01 ≈ 0.99999
    • x = -0.1 → sin(-0.1)/(-0.1) ≈ 0.9983
    • x = -0.01 → sin(-0.01)/(-0.01) ≈ 0.99999
  • Important caveat: Direct substitution x = 0 gives 0/0, so we cannot conclude the limit from that alone; a table or another method is needed.

One-sided limits

  • Motivation: when a function behaves differently on either side of a certain point, we study one-sided limits.
  • Example setup (piecewise function around x = 2):
    • As x → 2 from the left (x < 2): the outputs approach 4.
    • As x → 2 from the right (x > 2): the outputs approach 2.
  • Notation:
    • Left-hand limit: lim⁡x→2−f(x)=4.\lim_{x\to 2^-} f(x) = 4.
    • Right-hand limit: lim⁡x→2+f(x)=2.\lim_{x\to 2^+} f(x) = 2.
  • Conclusion: Since the left and right limits are not equal, the two-sided limit as x → 2 does not exist.
  • Relation to the graph:
    • The left side values near 2 lie on one branch of the graph (leading to 4).
    • The right side values near 2 lie on another branch (leading to 2).
    • The notation with minus and plus signs reflects approaching from the respective sides of the point.

Key takeaways and connections to broader concepts

  • A limit describes the behavior of f(x) as x approaches a, not necessarily the exact value at a.
  • A function can be undefined at a point (DNE at x = a for the function value) while still having a finite limit there.
  • The limit may exist (and be finite) even when the function value at that point is not defined or is different from the limit.
  • One-sided limits are essential when the function behaves differently on each side of a point; they tell us about the existence of the two-sided limit.
  • Special cases to watch for:
    • Indeterminate form 0/0 when directly substituting a; you must analyze further (simplify, use tables, or use other limit techniques).
    • Open circles indicate a hole (not in the domain) at that coordinate; a limit can exist despite the hole.
    • Closed circles indicate actual function values; the limit can differ from these values.
  • Real-world relevance: limits underpin the concepts of continuity, derivative definitions, and the behavior of functions near points of interest in modeling and analysis.

Quick recap of notation to remember

  • Two-sided limit: lim⁡x→af(x)=l.\lim_{x\to a} f(x) = l. (requires both sides to tend to l)
  • One-sided limits:
    • Left: lim⁡<em>x→a−f(x)=l</em>1\lim<em>{x\to a^-} f(x) = l</em>1
    • Right: lim⁡<em>x→a+f(x)=l</em>2\lim<em>{x\to a^+} f(x) = l</em>2
  • If $l1 = l2 = l$, then the two-sided limit exists and equals $l$.
  • If the function value at a differs or is undefined, the limit can still exist (or not) independently of the actual value at a.
  • Examples emphasized here:
    • $\lim_{x\to 0} g(x) = 2$ for $g(x) = \frac{x^2 + 2x}{x}$ with $x \neq 0$; $g(0)$ is undefined.
    • $\lim_{x\to 2} f(x) = 4$ for a piecewise function where $f(x) = x^2$ for $x \neq 2$ and $f(2) = 5$.
    • $\lim_{x\to 5} f(x) = 3$ even though $f(5) = 3$ (or undefined in the other example) depending on the case; a separate function value can differ from the limit.

Summary

  • Limits capture approaching values, not necessarily the actual value at the point.
  • Undefined points do not automatically make the limit undefined; whether the limit exists depends on the behavior near the point from both sides.
  • Distinguish between the function value at a point and the limit as x approaches that point.
  • Use both left and right limits to diagnose existence of the two-sided limit; one-sided limits may differ.
  • Practice with algebraic simplification, graphing intuition, and small-step table methods to estimate limits like in the sine example.