Limits are the foundation of calculus, allowing us to analyze function behavior near specific points. They help us understand continuity, rates of change, and function values at tricky spots.

Properties of limits simplify complex calculations by breaking them into manageable parts. We'll look at rules for basic operations, polynomials, powers, roots, and rational functions, as well as how to handle tricky situations like 0/0.

Properties of Limits

Limits of basic algebraic operations

• Sum Rule states the limit of a sum equals the sum of the limits ($\lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$)
  • Allows finding the limit of a sum by evaluating the limits of its individual components and adding the results
• Difference Rule states the limit of a difference equals the difference of the limits ($\lim_{x \to a} (f(x) - g(x)) = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$)
  • Enables finding the limit of a difference by evaluating the limits of its individual components and subtracting the results
• Product Rule states the limit of a product equals the product of the limits ($\lim_{x \to a} (f(x) \cdot g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$)
  • Allows finding the limit of a product by evaluating the limits of its individual components and multiplying the results
• Constant Multiple Rule states the limit of a constant multiple equals the constant multiple of the limit ($\lim_{x \to a} (c \cdot f(x)) = c \cdot \lim_{x \to a} f(x)$, where $c$ is a constant)
  • Enables finding the limit of a constant multiple by evaluating the limit of the function and multiplying it by the constant

Limits of polynomial functions

• Polynomial functions are continuous everywhere, meaning the limit of a polynomial function as $x$ approaches $a$ equals the value of the function at $x = a$
  • Continuity property simplifies the process of finding limits for polynomial functions
• Direct Substitution method finds $\lim_{x \to a} P(x)$, where $P(x)$ is a polynomial function, by evaluating $P(a)$
  • Plugging in the value of $a$ directly into the polynomial function yields the limit value
  • Example: For $P(x) = 3x^2 - 2x + 1$, $\lim_{x \to 2} P(x) = P(2) = 3(2)^2 - 2(2) + 1 = 9$

Limits with powers and roots

• Power Rule states the limit of a power equals the power of the limit ($\lim_{x \to a} (f(x))^n = (\lim_{x \to a} f(x))^n$, where $n$ is a real number)
  • Allows finding the limit of a power by evaluating the limit of the base function and raising it to the power
• Root Rule states the limit of an $n$th root equals the $n$th root of the limit ($\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}$, where $n$ is a positive integer)
  • Enables finding the limit of an $n$th root by evaluating the limit of the radicand and taking the $n$th root of the result
• Exponential Function Rule states the limit of an exponential function equals the exponential of the limit ($\lim_{x \to a} b^{f(x)} = b^{\lim_{x \to a} f(x)}$, where $b > 0$ and $b \neq 1$)
  • Allows finding the limit of an exponential function by evaluating the limit of the exponent and using it as the power of the base

Limits of rational functions

• Quotient Rule states the limit of a quotient equals the quotient of the limits, provided the limit of the denominator is not zero ($\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, where $\lim_{x \to a} g(x) \neq 0$)
  • Allows finding the limit of a quotient by evaluating the limits of the numerator and denominator separately and dividing the results
• Indeterminate Form $\frac{0}{0}$ occurs when both the numerator and denominator approach 0 as $x$ approaches $a$, indicating the limit may exist but requires further investigation
  • Factoring or canceling common factors can simplify the expression and help determine the limit value
  • Example: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} \frac{(x + 2)(x - 2)}{x - 2} = \lim_{x \to 2} (x + 2) = 4$
• Indeterminate Form $\frac{\infty}{\infty}$ occurs when both the numerator and denominator approach $\infty$ or $-\infty$ as $x$ approaches $a$, indicating the limit may exist but requires further investigation
  • Dividing both the numerator and denominator by the highest power of $x$ can help determine the limit value
  • Example: $\lim_{x \to \infty} \frac{3x^2 + 2x}{5x^2 - 1} = \lim_{x \to \infty} \frac{\frac{3x^2 + 2x}{x^2}}{\frac{5x^2 - 1}{x^2}} = \lim_{x \to \infty} \frac{3 + \frac{2}{x}}{5 - \frac{1}{x^2}} = \frac{3}{5}$

Continuity and Discontinuities

• Continuity is a fundamental concept in limit theory, describing functions that have no breaks or gaps
  • A function is continuous at a point if the limit exists and equals the function value at that point
• The epsilon-delta definition provides a precise mathematical description of continuity and limits
• Discontinuities can be classified into different types:
  • Removable discontinuity occurs when a function has a hole that can be "filled in" to make it continuous
  • Jump discontinuity happens when a function has a sudden change in value, creating a gap in its graph
• Asymptotic behavior describes how a function behaves as it approaches infinity or a vertical asymptote