Limits and Continuity Summary

Limits

• Definition: A function f has a limit L as x approaches a, written as \lim_{{x \to a}} f(x) = L, if f(x) approaches L as x approaches a from both sides.

• Left-hand limit: \lim_{{x \to a^-}} f(x) - Approaches a from the left side.

• Right-hand limit: \lim_{{x \to a^+}} f(x) - Approaches a from the right side.

• Two-sided limit: Both left-hand and right-hand limits exist and are equal: \lim_{{x \to a}} f(x) = b.

• Continuity: A function is continuous at x=a if:

  1. f(a) is defined
  2. \lim_{{x \to a}} f(x) exists
  3. \lim_{{x \to a}} f(x) = f(a)

• Discontinuities: Types include:

  • Jump: Different left-hand and right-hand limits exist.
  • Infinite: Vertical asymptote exists.
  • Removable: \lim_{{x \to a}} f(x) exists but f(a) is not equal to this limit.

Limit Properties

• 1. \lim_{{x \to a}} c = c (Constant)
• 2. \lim_{{x \to a}} x = a (Identity)
• 3. \lim{{x \to a}} [f(x) + g(x)] = \lim{{x \to a}} f(x) + \lim_{{x \to a}} g(x) (Sum)
• 4. \lim{{x \to a}} [f(x) - g(x)] = \lim{{x \to a}} f(x) - \lim_{{x \to a}} g(x) (Difference)
• 5. \lim{{x \to a}} [c \cdot f(x)] = c \cdot \lim{{x \to a}} f(x) (Constant Multiplier)
• 6. \lim{{x \to a}} [f(x) \cdot g(x)] = \lim{{x \to a}} f(x) \cdot \lim_{{x \to a}} g(x) (Product)
• 7. \lim{{x \to a}} \frac{f(x)}{g(x)} = \frac{\lim{{x \to a}} f(x)}{\lim_{{x \to a}} g(x)} (Quotient, g(a) ≠ 0)
• 8. \lim{{x \to a}} [f(x)]^n = (\lim{{x \to a}} f(x))^n (Power)
• 9. \lim{{x \to a}} \sqrt[n]{f(x)} = \sqrt[n]{\lim{{x \to a}} f(x)} (Root)

End Behaviour Examples

• Example with no limit: Sequence like 3^n - 2 does not approach a specific y-value.
• Example with limit: For \frac{x}{x+1} as x approaches infinity, limit approaches 1.

Graphing and Asymptotes

• Analyze graphs to evaluate limits at points of discontinuity and existence of asymptotes.

  • Example: If f(4) is undefined at a vertical asymptote, the function is discontinuous.

• Conditions for continuity at x=a must be checked for all three criteria outlined above.