Calculus: Limits and Continuity

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Limits and Continuity

• Limits describe the behavior of a function as it approaches a specific point.

The Idea of Limits

• The intuitive definition of a limit states:
• Assuming $f(x)$ is defined near $a$, then $\lim_{x \to a} f(x) = L$ if $f(x)$ gets arbitrarily close to $L$ as $x$ gets sufficiently close to $a$, but not equal to $a$.
• A precise definition of a limit is:
• Given a function $f$ defined on an open interval containing $a$, then $\lim_{x \to a} f(x) = L$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \epsilon$.

Limit Laws

• Assuming $c$ is a constant with the limits $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ existing:
• $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$ (Sum Law).
• $\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$ (Difference Law).
• $\lim_{x \to a} [cf(x)] = c \lim_{x \to a} f(x)$ (Constant Multiple Law).
• $\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$ (Product Law).
• $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$ if $\lim_{x \to a} g(x) \ne 0$ (Quotient Law).
• Direct Substitution Property:
• If $f$ is a polynomial or a rational function and $a$ is in the domain of $f$, then $\lim_{x \to a} f(x) = f(a)$.

Continuity

• A function $f$ is continuous at $a$ if $\lim_{x \to a} f(x) = f(a)$.
• If $f$ is continuous at $a$:
• $f(a)$ must be defined.
• $\lim_{x \to a} f(x)$ must exist.
• $\lim_{x \to a} f(x)$ must equal $f(a)$.
• A function $f$ is continuous from the right at $a$ if $\lim_{x \to a^+} f(x) = f(a)$.
• A function $f$ is continuous from the left at $a$ if $\lim_{x \to a^-} f(x) = f(a)$.