Calculus 1 (MAT 2003): Unit 1 - Limits and Continuity

These vocabulary flashcards cover the foundational concepts of limits and continuity based on the Calculus 1 (MAT 2003) lecture notes.

Calculus

The study of continuous change and a useful tool in the decision-making and planning process.

Function

A rule or relationship that assigns a unique value to each member or element of the domain (input values).

Real-valued functions

Functions where the domain and the co-domain are subsets of real numbers.

Limits

The building blocks of Calculus that describe or summarize the behavior of a function y-value as the x-value approaches a particular number.

Limit Notation

Expressed as $\lim_{x \to a} f(x) = L$, meaning as $x \to a$, then $f(x) \to L$, provided the limit exists.

Piecewise functions

Functions where the rule depends on the sub-domain.

Left hand Limit

A limit that focuses on the left-side of the x-value, $x = a$, denoted as $\lim_{x \to a^-} f(x)$, where x approaches a while increasing toward it.

Overall Limit Existence

The limit $\lim_{x \to a} f(x) = L$ exists if and only if both one-sided limits exist and are equal, such that $\lim_{x \to a^-} f(x) = \text{lim}_{x \to a^+} f(x)$.

Constant function limit rule

The limit of a constant function is simply the constant: $\lim_{x \to a} k = k$.

Limit of Sum/ Difference

The limit of a sum or difference is the sum or difference of the individual limits: \lim_{x \to a} (f(x) \text{ } \text{\pm} \text{ } g(x)) = \text{lim}_{x \to a} f(x) \text{ } \text{\pm} \text{ } \text{lim}_{x \to a} g(x).

Limit of a Product

The limit of a product is the product of the limits: $\lim_{x \to a} f(x)g(x) = \text{lim}_{x \to a} f(x) \times \text{lim}_{x \to a} g(x)$.

Indeterminate Form

A mathematical expression like $\frac{0}{0}$ where substituting values directly does not yield a specific limit; often addressed using the Factor/ Cancel approach.

Rational function

A function of the form $f(x) = \frac{N(x)}{D(x)}$ where $N(x)$ and $D(x)$ are polynomials.

Limit at infinity for Rational Functions

A rule stating that \lim_{x \to \text{\pm}\text{\infty}} \frac{k}{x^n} = 0, where $k$ is a constant and n \text{\ge} 1.

Continuity at a Point

A function is continuous at $x = a$ if $f(a)$ exists as a real number, $\lim_{x \to a} f(x)$ exists as a real number, and $\lim_{x \to a} f(x) = f(a)$.

Points of discontinuity (Rational Function)

A rational function $f(x) = \frac{P(x)}{Q(x)}$ is discontinuous when $Q(x) = 0$, resulting in the function being undefined at those points.