Section 10.2: More Limits (One-Sided, Infinite, and Limits at Infinity)

One-Sided Limits

  • Right-hand limit: lim⁡x→a+f(x)=L\lim_{x \to a^+} f(x) = L (approach from the right; x > a)
  • Left-hand limit: lim⁡x→a−f(x)=L′\lim_{x \to a^-} f(x) = L' (approach from the left; x < a)
  • Two-sided limit exists iff both one-sided limits exist and are equal: lim⁡<em>x→af(x)\lim<em>{x \to a} f(x) exists when lim⁡</em>x→a+f(x)=lim⁡x→a−f(x)\lim</em>{x \to a^+} f(x) = \lim_{x \to a^-} f(x)
  • Visual: if the two halves align (hole at a is allowed), the two-sided limit can exist; otherwise it does not.

Limits at Infinity vs Infinite Limits

  • Limits at infinity: values as x grows without bound (e.g., lim⁡<em>x→∞f(x)\lim<em>{x\to\infty} f(x) or lim⁡</em>x→−∞f(x)\lim</em>{x\to-\infty} f(x))
  • Infinite limits: limits that equal +∞+\infty or −∞-\infty as x approaches a from a side (not a finite number)
  • Distinction: "infinite limit" is about approaching ±∞ near a; "limit at infinity" is about the x variable going to ±∞ and the y-value approaching a finite number or ±∞.
  • Structure note: if the limit has a ratio with denominator going to zero and numerator not, you often get an infinite limit.

Example: infinite limit from a side

  • Consider lim⁡x→−2+xx+2\lim_{x \to -2^+} \frac{x}{x+2}
    • Numerator near −2-2 is negative; denominator x+2x+2 is positive (since x > -2 in the right-hand approach)
    • Overall sign is negative and magnitude grows without bound
    • Result: lim⁡x→−2+xx+2=−∞\lim_{x \to -2^+} \frac{x}{x+2} = -\infty

Limits at Infinity: leading-term principle

  • If f is rational (polynomial top and bottom), as x→∞x \to \infty, only the leading terms matter
  • Let numerator ~ axna x^n and denominator ~ bxmb x^m
    • If n < m: lim⁡x→∞axnbxm=0\lim_{x \to \infty} \frac{a x^n}{b x^m} = 0
    • If n=mn = m: lim⁡x→∞axnbxm=ab\lim_{x \to \infty} \frac{a x^n}{b x^m} = \frac{a}{b}
    • If n > m: lim⁡x→∞axnbxm=±∞\lim_{x \to \infty} \frac{a x^n}{b x^m} = \pm \infty (sign from a/ba/b)
  • Intuition: ignore lower-order terms as x becomes very large
  • Examples:
    • lim⁡<em>x→∞1−x3x2+1=lim⁡</em>x→∞−x3x2=−∞\lim<em>{x\to\infty} \frac{1 - x^3}{x^2 + 1} = \lim</em>{x\to\infty} \frac{-x^3}{x^2} = -\infty
    • If numerator degree < denominator degree, limit is 0

Vertical and Horizontal Asymptotes

  • Vertical asymptote at x = a when one-sided limits go to ±∞ near a:
    • E.g., left or right limit diverges
  • Horizontal asymptote y = L when lim⁡x→±∞f(x)=L\lim_{x\to\pm\infty} f(x) = L

Piecewise-Defined Functions and One-Sided Limits

  • When f is defined piecewise, determine the appropriate case by proximity to the limit point
  • Example:
    • f(x)=\begin{cases}
      \frac{1}{x}, & x
  • Limits:
    • lim⁡x→0+f(x)=3\lim_{x\to 0^+} f(x) = 3 (middle case)
    • lim⁡<em>x→1+f(x)=lim⁡</em>x→1+2−xx+1=12\lim<em>{x\to 1^+} f(x) = \lim</em>{x\to 1^+} \frac{2-x}{x+1} = \frac{1}{2}
    • lim⁡x→1−f(x)=3\lim_{x\to 1^-} f(x) = 3
  • Key idea: from the right, use the case that applies to numbers just larger than the target; from the left, use the case just smaller than the target.

Summary of Practical Rules

  • Two-sided limit exists iff left-hand and right-hand limits exist and are equal.
  • One-sided limits can exist even if the two-sided limit does not.
  • Infinite limits describe unbounded behavior near a; signs determine ±∞.
  • Limits at infinity are governed by leading terms in polynomials (or ratios): compare degrees and leading coefficients.
  • For piecewise functions, determine the relevant case by the side of approach.