Calculus 1 Review: Limits, Continuity, and Derivatives

Vocabulary and core theorems from calculus lecture notes covering limits, limit laws, continuity, derivative definitions, and differentiation rules.

Slope of the Secant Line

A measure of the slope of the tangent line, or the rate of change, of $f(x)$ at the given point $(a, f(a))$ calculated as $\frac{f(x) - f(a)}{x - a}$.

Instantaneous Velocity

The limiting values of the average velocities over shorter and shorter time periods.

Limit of a Function

To say that $\lim_{x \to a} f(x) = L$ means that as $x$ approaches $a$, but $x \neq a$, then $f(x)$ must approach $L$.

One-Sided Limits

A limit where the value is different when approaching from either the positive side ($x \to a^+$) or the negative side ($x \to a^-$).

Infinite Limit Theorem (Positive Even Integer)

If $n$ is a positive even integer, then $\lim_{x \to a^+} \frac{1}{(x-a)^n} = \infty$, $\lim_{x \to a^-} \frac{1}{(x-a)^n} = \infty$, and $\lim_{x \to a} \frac{1}{(x-a)^n} = \infty$.

Infinite Limit Theorem (Positive Odd Integer)

If $n$ is a positive odd integer, then $\lim_{x \to a^+} \frac{1}{(x-a)^n} = \infty$ and $\lim_{x \to a^-} \frac{1}{(x-a)^n} = -\infty$, hence the two-sided limit is DNE (Does Not Exist).

Sum Law

The limit law stating $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$.

Product Law

The limit law stating $\lim_{x \to a} [f(x) g(x)] = (\lim_{x \to a} f(x)) (\lim_{x \to a} g(x))$.

Direct Substitution Property

If $f$ is a function such that $a$ is in the domain of $f$, then $\lim_{x \to a} f(x) = f(a)$.

Squeeze Theorem

If $f(x) \leq g(x) \leq h(x)$ when $x$ is near $a$ (but not necessarily equal to $a$) and $\lim_{x \to a} f(x) = L = \lim_{x \to a} h(x)$, then $\lim_{x \to a} g(x) = L$.

Continuity at a Point

A function $f$ is continuous at $a$ if: 1. $f(a)$ is defined; 2. $\lim_{x \to a} f(x)$ exists; 3. $\lim_{x \to a} f(x) = f(a)$.

Limit Definition of Derivative

The derivative of a function at a number $a$, denoted by $f'(a)$, is defined as $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$.

Differentiable at a Number

A function $f$ is differentiable at a number $a$ if the limit $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ exists.

Theorem 3.1: Differentiability Implies Continuity

If $f(x)$ is differentiable at $a$, then $f$ is continuous at $a$.

The Power Rule

The differentiation rule stating that for any real number $n$, $\frac{d}{dx} [x^n] = n x^{n-1}$.

The Product Rule

The differentiation rule stating that $\frac{d}{dx} [f(x) g(x)] = f'(x) g(x) + f(x) g'(x)$.

The Quotient Rule

The differentiation rule stating that $\frac{d}{dx} [\frac{f(x)}{g(x)}] = \frac{g(x) f'(x) - f(x) g'(x)}{[g(x)]^2}$.

Particle at Rest

The state of a moving particle when its velocity at time $t$ is equal to zero ($v(t) = 0$).

Horizontal Tangent Line

A line on a function $f(x)$ where the slope is zero, found by setting the derivative $f'(x) = 0$.

Big Three Limits

1. $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$; 2. $\lim_{x \to 0} \frac{x}{\sin(x)} = 1$; 3. $\lim_{x \to 0} \frac{\tan(x)}{x} = 1$.