MTH 114 Calculus - Lecture 5: Calculus Foundations: Limits

Comprehensive vocabulary flashcards covering calculus limits, limit laws, indeterminate forms, one-sided limits, asymptotes, and evaluation techniques.

Limit of a Function ($\lim_{x \to a} f(x) = L$)

A mathematical statement describing that as the independent variable $x$ approaches $a$ (while $x \neq a$), the values of the function $f(x)$ get arbitrarily close to the real value $L$.

Graphical Behavior of Limits

The foundational principle that the limit of $f(x)$ as $x \to a$ depends exclusively on how $f(x)$ behaves near $x = a$, not at $x = a$ itself; thus, a limit can exist even if the function is undefined at $x = a$.

Two-Sided Limit Existence Theorem

The theorem establishing that the overall limit $\lim_{x \to a} f(x) = L$ exists if and only if both one-sided limits exist and are equal: $\lim_{x \to a^-} f(x) = L = \lim_{x \to a^+} f(x)$.

One-Sided Limit

A limit in which the variable $x$ approaches a specified number $a$ strictly from one direction: from the left ($x \to a^-$) or from the right ($x \to a^+$).

Contexts Requiring One-Sided Limits

The specific classes of functions that require evaluation via one-sided limits: piecewise functions, step functions, absolute value functions, rational functions with asymptotes, and functions containing breaks or undefined points.

Vertical Asymptote

A vertical line $x = a$ corresponding to an infinite limit where $f(x)$ increases without bound ($f(x) \to \infty$) or decreases without bound ($f(x) \to -\infty$) as $x$ approaches $a$ from either the left or the right.

Horizontal Asymptote

A horizontal line $y = L$ where the values of $f(x)$ approach the finite constant $L$ as the variable $x$ approaches positive infinity ($x \to +\infty$) or negative infinity ($x \to -\infty$).

Direct Substitution

The preliminary method of evaluating a limit $\lim_{x \to a} f(x)$ by calculating $f(a)$; if this yields a finite real number $b$, then the limit is finite and equal to $b$.

Indeterminate Forms

Mathematical expressions resulting from direct substitution whose limits cannot be directly established without algebraic transformation or L'Hôpital's Rule: $\frac{0}{0}$, $\frac{\pm\infty}{\pm\infty}$, $\infty - \infty$, $0 \times \infty$, $\infty^0$, $1^\infty$, and $0^0$.

Sum and Difference Rules for Limits

The limit laws asserting that if $\lim_{x \to a} f(x) = L_1$ and $\lim_{x \to a} g(x) = L_2$, then $\lim_{x \to a} [f(x) + g(x)] = L_1 + L_2$ and $\lim_{x \to a} [f(x) - g(x)] = L_1 - L_2$.

Product Rule for Limits

The limit law asserting that if $\lim_{x \to a} f(x) = L_1$ and $\lim_{x \to a} g(x) = L_2$, then the limit of their product is the product of their limits: $\lim_{x \to a} [f(x)g(x)] = L_1 L_2$.

Quotient Rule for Limits

The limit law stating that if $\lim_{x \to a} f(x) = L_1$ and $\lim_{x \to a} g(x) = L_2$, then $\lim_{x \to a} \left[\frac{f(x)}{g(x)}\right] = \frac{L_1}{L_2}$, provided that the denominator limit satisfies $L_2 \neq 0$.

Root Rule for Limits

The limit law stating that $\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{L_1}$, where $L_1 > 0$ is required whenever the integer root index $n$ is even.

Decomposition (Factorization) Method

An algebraic technique applied to resolve $\frac{0}{0}$ indeterminate quotients by factoring the polynomials in the numerator and denominator, canceling common factors, and re-evaluating the limit.

Conjugate Multiplication Method

An algebraic technique used to resolve indeterminate forms of type $\frac{0}{0}$ involving radical expressions by multiplying both numerator and denominator by the conjugate of the radical expression.

Special Trigonometric Limits: Sine

The standard trigonometric limit identities stating that $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$, $\lim_{x \to 0} \frac{\sin(ax)}{x} = a$, $\lim_{x \to 0} \frac{x}{\sin(x)} = 1$, and $\lim_{x \to 0} \frac{x}{\sin(ax)} = \frac{1}{a}$.

Special Trigonometric Limits: Tangent

The standard trigonometric limit identities stating that $\lim_{x \to 0} \frac{\tan(x)}{x} = 1$, $\lim_{x \to 0} \frac{\tan(ax)}{x} = a$, $\lim_{x \to 0} \frac{x}{\tan(x)} = 1$, and $\lim_{x \to 0} \frac{x}{\tan(ax)} = \frac{1}{a}$.

Special Trigonometric Limit: Cosine

The standard trigonometric limit identity stating that $\lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0$.

End Behavior of Sine ($\sin(x)$)

The asymptotic characteristic of $y = \sin(x)$ oscillating perpetually between $-1$ and $1$, which causes $\lim_{x \to +\infty} \sin(x)$ and $\lim_{x \to -\infty} \sin(x)$ not to exist.

End Behavior of Exponential Functions ($e^x$)

The asymptotic properties of the natural exponential function defined by $\lim_{x \to +\infty} e^x = +\infty$ and $\lim_{x \to -\infty} e^x = 0$.

End Behavior of Natural Logarithmic Function ($\ln(x)$)

The asymptotic behavior of the natural logarithmic function defined by $\lim_{x \to +\infty} \ln(x) = +\infty$ and the right-sided vertical asymptote behavior $\lim_{x \to 0^+} \ln(x) = -\infty$.

Rational Functions Limit at Infinity Method

The analytical procedure for evaluating $\lim_{x \to \pm\infty} \frac{P(x)}{Q(x)}$ by dividing every term in both the numerator and denominator by $x^k$, where $k$ is the highest power of $x$ present in the denominator.