Calculus I: Limits, Continuity, and Derivatives

Vocabulary practice flashcards covering core definitions, limit theorems, continuity classifications, and derivative rules from Math 124 lecture notes.

Secant line

A line passing through two distinct points $P$ and $Q$ on a curve $C$.

Tangent line

The line through a point $P(x_1, y_1)$ on a curve $C$ of equation $y = f(x)$ with slope $m = \lim_{x \to x_1} \frac{f(x) - f(x_1)}{x - x_1}$.

Derivative of $f$ at $x_1$

The limit $f'(x_1) = \lim_{x \to x_1} \frac{f(x) - f(x_1)}{x - x_1}$ (or equivalently $\lim_{h \to 0} \frac{f(x_1 + h) - f(x_1)}{h}$), which represents the slope of the tangent line to $y = f(x)$ at $(x_1, f(x_1))$ and the instantaneous rate of change of $f$ at $x_1$.

Differentiable at $x_1$

A function $f$ is differentiable at $x_1$ if the derivative limit $\lim_{x \to x_1} \frac{f(x) - f(x_1)}{x - x_1}$ exists.

Average rate of change

The ratio $\frac{f(t_2) - f(t_1)}{t_2 - t_1}$ over an interval $[t_1, t_2]$, which equals the slope of the secant line between the points on the graph of $f$.

Instantaneous rate of change

The rate of change of $f$ at an exact point $x_1$, defined by the derivative $f'(x_1)$ (the limit of the average rate of change as $t_2$ gets very close to $t_1$).

Limit from the right

The value that $f(x)$ approaches as $x$ gets closer and closer to $x_1$ while remaining strictly greater than $x_1$, denoted $\lim_{x \to x_1^+} f(x)$.

Limit from the left

The value that $f(x)$ approaches as $x$ gets closer and closer to $x_1$ while remaining strictly less than $x_1$, denoted $\lim_{x \to x_1^-} f(x)$.

Continuous at a point

A function $f$ is continuous at a point $x_0$ if $f$ is defined at $x_0$ and $\lim_{x \to x_0} f(x) = f(x_0)$.

Continuous from the right

A function $f$ is continuous at $x_0$ from the right if $f$ is defined at $x_0$ and $\lim_{x \to x_0^+} f(x) = f(x_0)$.

Continuous from the left

A function $f$ is continuous at $x_0$ from the left if $f$ is defined at $x_0$ and $\lim_{x \to x_0^-} f(x) = f(x_0)$.

Removable discontinuity

A discontinuity at a point $x_0$ in the domain of $f$ where both $f(x_0)$ and $\lim_{x \to x_0} f(x) = L$ exist, but $f(x_0) \neq L$.

Jump discontinuity

A discontinuity at a point $x_0$ in the domain of $f$ where both one-sided limits exist and are finite ($\lim_{x \to x_0^-} f(x) = L_1$ and $\lim_{x \to x_0^+} f(x) = L_2$), but $L_1 \neq L_2$.

Infinite discontinuity

A discontinuity at a point $x_0$ in the domain of $f$ where one or both of the one-sided limits $\lim_{x \to x_0^-} f(x)$ or $\lim_{x \to x_0^+} f(x)$ is infinite or does not exist.

Squeeze theorem

If $g(x) \le f(x) \le h(x)$ for all $x$ close to $x_0$ and $\lim_{x \to x_0} g(x) = \lim_{x \to x_0} h(x) = L$, then $\lim_{x \to x_0} f(x) = L$.

Horizontal asymptote

The line $y = L$ is a horizontal asymptote for $y = f(x)$ if either $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$.

Vertical asymptote

The line $x = a$ is a vertical asymptote for $y = f(x)$ if either $\lim_{x \to a^+} f(x) = \pm\infty$ or $\lim_{x \to a^-} f(x) = \pm\infty$.

End-behavior rule for rational functions

For rational functions, $\lim_{x \to \pm\infty} \frac{a_n x^n + \dots}{b_m x^m + \dots}$ behaves like $\lim_{x \to \pm\infty} \frac{a_n x^n}{b_m x^m}$, disregarding lower-degree terms.

Elementary functions

The collection of functions consisting of constants, powers ($x^n$), roots ($\sqrt[m]{x}$), exponentials ($a^x$), logarithms ($\ln(x)$), and trigonometric and inverse trigonometric functions.

Indeterminate forms

Limit expressions whose values cannot be determined without further algebraic manipulation, including $\frac{0}{0}$, $\frac{\infty}{\infty}$, $0 \cdot (\infty)$, and $(+\infty) + (-\infty)$.

Fundamental trigonometric limit for sine

The limit identity stating that $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$, and more generally $\lim_{x \to a} \frac{\sin(f(x))}{f(x)} = 1$ if $\lim_{x \to a} f(x) = 0$.

Product rule

The differentiation rule for the product of two differentiable functions: $(fg)' = f'g + fg'$.

Quotient rule

The differentiation rule for the quotient of two differentiable functions: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$.

Non-differentiability conditions

The three graphical conditions under which a function $f$ is not differentiable at $x$: (1) $f$ is not continuous at $x$, (2) $f$ has a sharp corner at $x$, or (3) $f$ has a vertical tangent line at $x$.

External tangent line method

A technique to find a tangent line to $y = f(x)$ passing through an external point $P(x_0, y_0)$ by equating the derivative to the secant slope expression, $f'(x) = \frac{f(x) - y_0}{x - x_0}$, and solving for the point of tangency $x$.

Rationalizing technique

An algebraic method used to evaluate limits resulting in $\frac{0}{0}$ by multiplying the numerator and denominator by the conjugate of an expression involving radicals.