Limit laws are powerful tools for breaking down complex limits into simpler parts. These laws allow us to add, subtract, multiply, and divide limits, making it easier to evaluate tricky expressions. They're like a Swiss Army knife for calculus problems.

Mastering these techniques opens up a world of problem-solving possibilities. From polynomials to rational functions, we can tackle a wide range of limit problems. The squeeze theorem and l'Hôpital's rule are especially handy for those head-scratching situations where direct substitution fails us.

Limit Laws and Techniques

Fundamental limit laws

• Break down complex limits into smaller, more manageable parts using limit laws
  • Sum law adds the limits of two functions $f(x)$ and $g(x)$ as $x$ approaches $a$: $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$
  • Difference law subtracts the limits of two functions $f(x)$ and $g(x)$ as $x$ approaches $a$: $\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$
  • Product law multiplies the limits of two functions $f(x)$ and $g(x)$ as $x$ approaches $a$: $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
  • Quotient law divides the limits of two functions $f(x)$ and $g(x)$ as $x$ approaches $a$, provided that $\lim_{x \to a} g(x) \neq 0$: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$
  • Power law raises the limit of a function $f(x)$ to the power $n$ as $x$ approaches $a$, where $n$ is a positive integer: $\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$
  • Root law takes the $n$th root of the limit of a function $f(x)$ as $x$ approaches $a$, where $n$ is a positive integer: $\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}$
• Constant multiple rule multiplies the limit of a function $f(x)$ by a constant $c$ as $x$ approaches $a$: $\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)$
• Function composition law states that if $\lim_{x \to a} g(x) = L$ and $f$ is continuous at $L$, then $\lim_{x \to a} f(g(x)) = f(\lim_{x \to a} g(x)) = f(L)$

Limits of polynomial functions

• Evaluate limits of polynomial functions by substituting the value of $a$ directly into the function
  • $\lim_{x \to 2} (3x^2 - 4x + 1) = 3(2)^2 - 4(2) + 1 = 5$
• Evaluate limits of rational functions by substituting the value of $a$ directly into the function, as long as the denominator is not zero
  • $\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = \frac{1^2 - 1}{1 - 1} = \frac{0}{0}$ (indeterminate form)
    • Simplify indeterminate forms using factoring or other algebraic techniques

Simplifying complex limit expressions

• Simplify rational functions by factoring the numerator and denominator and canceling common factors
  • $\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = \lim_{x \to 1} \frac{(x - 1)(x + 1)}{x - 1} = \lim_{x \to 1} (x + 1) = 2$
• Simplify expressions containing square roots by multiplying the numerator and denominator by the conjugate of the numerator
  • $\lim_{x \to 0} \frac{\sqrt{x + 1} - 1}{x} = \lim_{x \to 0} \frac{(\sqrt{x + 1} - 1)(\sqrt{x + 1} + 1)}{x(\sqrt{x + 1} + 1)} = \lim_{x \to 0} \frac{x}{x(\sqrt{x + 1} + 1)} = \lim_{x \to 0} \frac{1}{\sqrt{x + 1} + 1} = \frac{1}{2}$
• Apply l'Hôpital's rule for indeterminate forms of type 0/0 or ∞/∞ by taking the derivative of both numerator and denominator

Application of squeeze theorem

• Apply the squeeze theorem (sandwich theorem) when a function $f(x)$ is bounded between two functions $g(x)$ and $h(x)$ for all $x$ near $a$ (except possibly at $a$), and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$
  • $0 \leq |\sin x| \leq |x|$ for all $x$, and $\lim_{x \to 0} 0 = \lim_{x \to 0} |x| = 0$, so $\lim_{x \to 0} |\sin x| = 0$

Specific limit laws for functions

• Polynomial functions use direct substitution
• Rational functions use direct substitution, factoring, or conjugates
• Trigonometric functions use the squeeze theorem or trigonometric identities
• Exponential and logarithmic functions use properties of exponents and logarithms

Function behavior near points

• Evaluate one-sided limits of a function as $x$ approaches $a$ from the left ($x \to a^-$) and right ($x \to a^+$) to determine the behavior of the function near $a$
  • If $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$, the two-sided limit exists and equals the common value
  • If $\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$, the two-sided limit does not exist, indicating a discontinuity at $x = a$
• Determine the behavior of a function as $x$ approaches $a$ when the limit approaches infinity or negative infinity (infinite limits)
  • $\lim_{x \to 0^+} \frac{1}{x} = \infty$ and $\lim_{x \to 0^-} \frac{1}{x} = -\infty$, indicating a vertical asymptote at $x = 0$

Continuity and Limit Definitions

• Understand the epsilon-delta definition of a limit to formally prove limit statements
• Use the concept of continuity to determine if a function is continuous at a point or on an interval
• Apply the intermediate value theorem to prove the existence of solutions to equations on a closed interval