Calculus: Understanding Limits

• Calculus is a branch of mathematics that deals with continuous change.
• It provides tools and techniques to study rates of change and accumulation.
• Limits are a foundational concept in calculus, providing the basis for defining continuity, derivatives, and integrals.

Intuitive Understanding of Limits

• The limit of a function describes the value that a function approaches as its input (or argument) gets arbitrarily close to a particular value.
• It focuses on the behavior of the function near a point, rather than exactly at that point.
• lim (x→c) f(x) = L means "the limit of f(x) as x approaches c is equal to L".
  • x→c indicates that x approaches c, but does not necessarily equal c.
  • L is the limiting value.
• The limit exists if the function approaches the same value from both the left and the right side of c.

Formal Definition of a Limit (ε-δ Definition)

• For every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε.
  • ε represents an arbitrarily small positive number that defines how close f(x) needs to be to L.
  • δ represents an arbitrarily small positive number that defines how close x needs to be to c.
  • |x - c| < δ means x is within a distance of δ from c, but not equal to c (due to 0 < |x - c|).
  • |f(x) - L| < ε means f(x) is within a distance of ε from L.
• The definition formalizes approaching, by stating that f(x) can be made arbitrarily close to L by making x sufficiently close to c.

One-Sided Limits

• Sometimes, the limit of a function as x approaches c exists only from one side.
• The limit as x approaches c from the right (x > c) is denoted as lim (x→c+) f(x).
• The limit as x approaches c from the left (x < c) is denoted as lim (x→c-) f(x).
• For a two-sided limit to exist (lim (x→c) f(x)), both one-sided limits must exist and be equal: lim (x→c-) f(x) = lim (x→c+) f(x).

Techniques for Evaluating Limits

• Direct Substitution: If f(x) is continuous at x = c, then lim (x→c) f(x) = f(c).
• Factoring: Factor the expression to simplify and cancel out terms that cause indeterminacy (e.g., 0/0).
• Rationalizing: Multiply the numerator and denominator by the conjugate to eliminate radicals and simplify.
• Trigonometric Identities: Use trigonometric identities to rewrite the expression in a more manageable form.
• L'Hôpital's Rule: If the limit results in an indeterminate form (0/0 or ∞/∞), take the derivative of the numerator and denominator separately and then evaluate the limit.
  • lim (x→c) f(x) / g(x) = lim (x→c) f'(x) / g'(x), provided the limit on the right exists.
• Squeeze (Sandwich) Theorem: If g(x) ≤ f(x) ≤ h(x) for all x near c (except possibly at c) and lim (x→c) g(x) = lim (x→c) h(x) = L, then lim (x→c) f(x) = L.

Limit Laws

• Sum/Difference Law: lim (x→c) [f(x) ± g(x)] = lim (x→c) f(x) ± lim (x→c) g(x)
• Constant Multiple Law: lim (x→c) [k * f(x)] = k * lim (x→c) f(x), where k is a constant.
• Product Law: lim (x→c) [f(x) * g(x)] = lim (x→c) f(x) * lim (x→c) g(x)
• Quotient Law: lim (x→c) [f(x) / g(x)] = lim (x→c) f(x) / lim (x→c) g(x), provided lim (x→c) g(x) ≠ 0.
• Power Law: lim (x→c) [f(x)]^n = [lim (x→c) f(x)]^n, where n is a real number.

Limits Involving Infinity

• Limits at Infinity: Evaluate the behavior of a function as x approaches positive or negative infinity (lim (x→∞) f(x) or lim (x→-∞) f(x)).
• Infinite Limits: The limit of a function is infinite (positive or negative) as x approaches a particular value (lim (x→c) f(x) = ∞ or lim (x→c) f(x) = -∞).
• Vertical Asymptotes: If lim (x→c) f(x) = ±∞, then x = c is a vertical asymptote of f(x).
• Horizontal Asymptotes: If lim (x→∞) f(x) = L or lim (x→-∞) f(x) = L, then y = L is a horizontal asymptote of f(x).

Indeterminate Forms

• Indeterminate forms are expressions that do not have a definite value and require further analysis to evaluate the limit.
• Common indeterminate forms include: 0/0, ∞/∞, 0 * ∞, ∞ - ∞, 1^∞, 0^0, ∞^0.
• L'Hôpital's Rule is often used to evaluate limits with indeterminate forms 0/0 or ∞/∞.

Continuity

• A function f(x) is continuous at x = c if the following three conditions are met:
  • f(c) is defined (c is in the domain of f).
  • lim (x→c) f(x) exists.
  • lim (x→c) f(x) = f(c).
• A function is continuous on an interval if it is continuous at every point in the interval.
• Types of Discontinuities:
  • Removable Discontinuity: The limit exists, but it's not equal to the function value at that point.
  • Jump Discontinuity: The left and right limits exist but are not equal.
  • Infinite Discontinuity: The function approaches infinity at that point.
  • Oscillating Discontinuity: The function oscillates infinitely many times near the point.