Limits at Infinity and Average Rate of Change

Limits at Infinity for Rational Functions

• When x → ±∞, both numerator and denominator polynomials grow large; ∞/∞ is an indeterminate form and must be evaluated.

• Long method (for a rational function f(x) = P(x)/Q(x)): divide every term by the highest power of x that appears in the denominator.

  • If the denominator has degree n, divide by x^n and simplify.
  • Terms like 1/x^k go to 0 as x → ∞.

• Shortcut rules (based on degrees):

  • Case 1: deg(top) < deg(bottom) → limit = 0.
  • Case 2: deg(top) > deg(bottom) → limit does not exist (diverges to ±∞).
  • Case 3: deg(top) = deg(bottom) → limit equals the ratio of the leading coefficients of the highest-degree terms.

• Example intuition: numerator grows slower than denominator → ratio tends to 0; numerator grows faster → ratio diverges.

• Non-zero over zero form at a finite limit point: If
  rac{P(x)}{Q(x)}
  has Q(c) = 0 but P(c) ≠ 0, the limit as x → c does not exist (it tends to ±∞; one-sided limits may differ).

  • Example form: ext{lim}_{x o 5} rac{3x^2 + 1}{x - 5}
    as x → 5, numerator → 76 (nonzero) and denominator → 0, so the limit is not finite; right-hand limit → +∞, left-hand limit → −∞.

• Average rate of change (section 11.3):

  • Definition: average rate of change of f from a to b is
    rac{f(b) - f(a)}{b - a}
  • Graphically: slope of the secant line through (a, f(a)) and (b, f(b)).
  • Secant line vs tangent line:
  • Secant line: line through two points on the graph.
  • Tangent line: line touching the graph at exactly one point.

• Example 1: Average rate of change of f(x) = 3x^2 + 2 from x = 1 to x = 4

  • f(4) = 3(4^2) + 2 = 50
  • f(1) = 3(1^2) + 2 = 5
  • Change in f: f(4) - f(1) = 50 - 5 = 45
  • Change in x: 4 - 1 = 3
  • Average rate of change: rac{45}{3} = 15
  • This is the slope of the secant line through (1, f(1)) and (4, f(4)).

• Example 2: Average velocity from t = 2 to t = 4 for position s(t) = t^2 + 3t - 2

  • s(4) = 4^2 + 3(4) - 2 = 26
  • s(2) = 2^2 + 3(2) - 2 = 8
  • Time elapsed: 4 − 2 = 2
  • Distance traveled: s(4) - s(2) = 26 - 8 = 18
  • Average velocity: rac{18}{2} = 9 ext{ ft/s}
  • Interpretation: average rate of change of position equals average velocity over the interval.

• Quick recap:

  • If deg(top) < deg(bottom): limit → 0 as x → ±∞.
  • If deg(top) > deg(bottom): limit does not exist (diverges to ±∞).
  • If deg(top) = deg(bottom): limit = ratio of leading coefficients.
  • For finite x → c with Q(c) = 0 and P(c) ≠ 0: limit does not exist (infinite behavior).
  • Average rate of change equals the slope of the secant line; connects endpoint values of the function on an interval.