Calculus: Limits and Continuity Concepts

Limits

• Definition: The value that a function approaches as the input approaches a certain value.
• Notation: (\lim_{x \to c} f(x) = L) means as (x) approaches (c), (f(x)) approaches (L).

Continuity

• Definition: A function (f(x)) is continuous at (x = c) if:
  1. (f(c)) is defined.
  2. (\lim_{x \to c} f(x)) exists.
  3. (\lim_{x \to c} f(x) = f(c)).

Function Behavior

• Increasing/Decreasing: Look for intervals where the derivative (f'(x) > 0) (increasing) or (f'(x) < 0) (decreasing).
• Local Extrema: Use first derivative test to find local maximum and minimum points.
• End Behavior: Analyze limits as (x \to \infty) or (x \to -\infty).

Rate of Change

• Average Rate of Change: Given two points (a) and (b) on the function (f), the average rate of change is (\frac{f(b) - f(a)}{b - a}).
• Instantaneous Rate of Change: The derivative (f'(x)) provides the instantaneous rate of change at a point.

Squeeze Theorem

• Concept: If (g(x) \leq f(x) \leq h(x)) and (\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L), then (\lim_{x \to c} f(x) = L).

Intermediate Value Theorem (IVT)

• Statement: If (f) is continuous on ([a, b]) and (N) is a value between (f(a)) and (f(b)), there exists at least one (c \in (a, b)) such that (f(c) = N).

Determining Limits Algebraically

• Direct Substitution: Start by substituting the value into the function.
• Factoring: Factor expressions to identify removable discontinuities.
• Rationalization: Use for limits involving roots.
• L'Hôpital's Rule: Apply when encountering indeterminate forms (0/0 or ∞/∞).

Continuities Algebraically

• Piecewise Functions: Check continuity at the points where the pieces meet.
• Rational Functions: Identify points of discontinuity where the denominator is zero.
• Trigonometric Functions: Continuous everywhere unless there are restrictions in the domain.

Infinite Limits and Asymptotes

• Infinite Limits: Occur when (\lim_{x \to c} f(x) = \infty):
  • Indicates a vertical asymptote at (x = c).
• Vertical Asymptotes: Found where the function approaches infinity; typically where the denominator is zero.
• Horizontal Asymptotes: Determined by evaluating (\lim_{x \to \infty} f(x)) or (\lim_{x \to -\infty} f(x)):
  • If the limit approaches a constant (L), (y = L) is a horizontal asymptote.

Limits

• A limit represents the value a function approaches as the input nears a specific number.
• Notation: (\lim_{x \to c} f(x) = L) indicates that as (x) approaches (c), (f(x)) approaches (L).

Continuity

• A function (f(x)) is considered continuous at (x = c) if:
  • (f(c)) is defined.
  • The limit (\lim_{x \to c} f(x)) exists.
  • The limit equals the function value: (\lim_{x \to c} f(x) = f(c)).

Function Behavior

• Increasing/Decreasing:
  • A function is increasing where its derivative, (f'(x)), is positive.
  • It is decreasing where (f'(x)) is negative.
• Local Extrema:
  • Determine local maxima and minima using the first derivative test.
• End Behavior:
  • Examine limits as (x) tends to infinity ((x \to \infty)) or negative infinity ((x \to -\infty)) to understand long-term trends of a function.

Rate of Change

• Average Rate of Change:
  • Calculated between two points (a) and (b) on the function as (\frac{f(b) - f(a)}{b - a}).
• Instantaneous Rate of Change:
  • The derivative (f'(x)) indicates the rate of change at a particular point.

Squeeze Theorem

• The theorem states that if (g(x) \leq f(x) \leq h(x)) and both limits (\lim_{x \to c} g(x)) and (\lim_{x \to c} h(x)) equal (L), then (\lim_{x \to c} f(x)) also equals (L).

Intermediate Value Theorem (IVT)

• If (f) is continuous on the interval ([a, b]) and (N) lies between (f(a)) and (f(b)), then there exists at least one (c) in ((a, b)) such that (f(c) = N).

Determining Limits Algebraically

• Direct Substitution: Substitute the specific value into the function directly.
• Factoring: Solve by factoring to find and eliminate removable discontinuities.
• Rationalization: Apply when limits involve square roots.
• L'Hôpital's Rule: Use for indeterminate forms such as (0/0) or (\infty/\infty).

Continuities Algebraically

• Piecewise Functions: Assess continuity at transition points between pieces.
• Rational Functions: Identify discontinuity points where the denominator becomes zero.
• Trigonometric Functions: Generally continuous unless domain constraints exist.

Infinite Limits and Asymptotes

• Infinite Limits: Occur when (\lim_{x \to c} f(x) = \infty), indicating a vertical asymptote at (x = c).
• Vertical Asymptotes: Found where the function approaches infinity, often linked to points where the denominator is zero.
• Horizontal Asymptotes: Established by evaluating (\lim_{x \to \infty} f(x)) or (\lim_{x \to -\infty} f(x)):
  • If the limit approaches a constant (L), then (y = L) serves as a horizontal asymptote.

Limits

• Definition: A limit describes the value that a function approaches as the input approaches some value.
• Notation: The limit of f(x) as x approaches a is expressed as lim (x→a) f(x).
• One-sided limits: Left-hand limit (lim (x→a⁻) f(x)) and right-hand limit (lim (x→a⁺) f(x)).
• Infinite limits: When f(x) approaches infinity as x approaches a particular value, indicating vertical asymptotes.

Continuity

• A function is continuous at a point if:
  • The limit as x approaches that point exists.
  • The function value at that point is defined.
  • The function value matches the limit at that point.
• Types of discontinuities:
  • Removable: The limit exists, but the function is not defined at that point.
  • Jump: The left-hand and right-hand limits exist, but are not equal.
  • Infinite: The function approaches infinity at a point.

Intermediate Value Theorem (IVT)

• Stipulates that for any value between f(a) and f(b), there exists at least one c in (a, b) such that f(c) equals that value.
• Applicable only if f is continuous on the interval [a, b].

Squeeze Theorem

• Used to determine the limit of a function that is "squeezed" between two other functions.
• If g(x) ≤ f(x) ≤ h(x) for all x in an interval (except possibly at c), and lim (x→c) g(x) = lim (x→c) h(x) = L, then lim (x→c) f(x) = L.

Infinite Limits

• Occurs when the values of f(x) approach infinity as x approaches a specific point.
• Leads to vertical asymptotes on the graph of the function.

Horizontal Asymptotes

• A horizontal line that the graph of a function approaches as x approaches infinity or negative infinity.
• Determined by comparing the degrees of the numerator and denominator in rational functions:
  • Degree of numerator < degree of denominator: y = 0 is a horizontal asymptote.
  • Degree of numerator = degree of denominator: y = leading coefficient of numerator/leading coefficient of denominator.
  • Degree of numerator > degree of denominator: no horizontal asymptote (may have an oblique asymptote).

Vertical Asymptotes

• Occur at points where a function approaches infinity or negative infinity.
• Identified by values of x that make the denominator zero while the numerator is non-zero.