Functions, Limits, and Continuity

Functions, Limits, and Continuity

Differential calculus begins with a rigorous understanding of functions and their behavior as inputs approach specific values. This topic covers the essential building blocks: limits and continuity.

Limit Laws

If $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$, then:

1. Sum/Difference: $\lim_{x \to c} [f(x) \pm g(x)] = L \pm M$
2. Product: $\lim_{x \to c} [f(x) \cdot g(x)] = L \cdot M$
3. Quotient: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L}{M}$ (provided $M \neq 0$)
4. Power: $\lim_{x \to c} [f(x)]^n = L^n$
5. Root: $\lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{L}$

Continuity

A function $f(x)$ is continuous at a point $x = c$ if three conditions are met:

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

Infinite Limits and Limits at Infinity

Infinite Limits: If $f(x)$ increases or decreases without bound as $x \to c$, we say the limit is $\infty$ or $-\infty$. This indicates a vertical asymptote at $x = c$.

Limits at Infinity: We analyze the behavior of $f(x)$ as $x \to \infty$ or $x \to -\infty$.

• For rational functions $\frac{P(x)}{Q(x)}$:
  • If degree of $P <$ degree of $Q$, limit is 0.
  • If degree of $P >$ degree of $Q$, limit is $\pm \infty$.
  • If degree of $P =$ degree of $Q$, limit is the ratio of leading coefficients.