Infinite Limits, Horizontal Asymptotes, and Continuity in Calculus

Infinite Limits and Horizontal Asymptotes

Definition of Horizontal Asymptote

A function f(x) has a horizontal asymptote at y = L if either of the following is true:

• $ext{lim}_{x o ext{infinity}} f(x) = L$

• $ext{lim}_{x o - ext{infinity}} f(x) = L$

This means that as x becomes very large (positively or negatively), the function approaches the constant value L.

Examples of Horizontal Asymptotes

• For $f(x) = rac{5x^2 + 4}{3x^2 - x}$, as $x o ext{infinity}$, the leading terms dominate, so $f(x) o rac{5}{3}$.

• For $f(x) = rac{4x + 1}{3x - 5}$, as $x o ext{infinity}$, $f(x) o rac{4}{3}$.

Key Point: For rational functions, the horizontal asymptote is determined by the ratio of the leading coefficients if the degrees of numerator and denominator are equal.

Behavior as $x o ext{infinity}$

• For $f(x) = x^2 - x$, as $x o ext{infinity}$, $f(x) o ext{infinity}$.

• For $f(x) = rac{2x + 3}{5x + 1}$, as $x o ext{infinity}$, $f(x) o rac{2}{5}$.

• For $f(x) = ext{ln}(2x + 3) - ext{ln}(5x + 1)$, as $x o ext{infinity}$, $f(x) o ext{ln}(2/5)$.

Example: $\lim_{x \to \infty} \text{ln}(2x + 3) - \text{ln}(5x + 1) = \text{ln}(2/5)$

Oscillatory Limits and Non-Existence

Oscillatory Functions

Functions such as $\sin(x)$, $\cos(x)$, and $\tan(x)$ do not approach a single value as $x \to \infty$ or $x \to -\infty$ due to their periodic nature. Their limits do not exist (DNE) in these cases.

• $\lim_{x \to \infty} \sin(x)$ DNE

• $\lim_{x \to \infty} \cos(x)$ DNE

• $\lim_{x \to \infty} \tan(x)$ DNE

Key Point: Oscillatory or periodic functions do not settle to a single value at infinity.

Direct Substitution Law for Limits

Polynomial and Rational Functions

• If $f(x)$ is a polynomial or rational function and $a$ is in its domain, then $\lim_{x \to a} f(x) = f(a)$.

Trigonometric Functions

• If $g(x)$ is a trigonometric function and $a$ is in its domain, then $\lim_{x \to a} g(x) = g(a)$.

Key Point: Direct substitution works for continuous functions at points in their domain.

Continuity

Definition of Continuity at a Point

A function f(x) is continuous at a number a if:

• $f(a)$ is defined

• $\lim_{x \to a} f(x)$ exists

• $\lim_{x \to a} f(x) = f(a)$

If any of these conditions fail, the function has a discontinuity at a.

Types of Discontinuities

• Removable Discontinuity: The limit exists, but $f(a)$ is not defined or does not match the limit.

• Jump Discontinuity: The left and right limits exist but are not equal.

• Infinite/Essential Discontinuity: The limit does not exist because the function approaches infinity or oscillates without bound.

Examples of Discontinuities

• For $f(x) = \frac{x^2 - 1}{x - 1}$ if $x \neq 1$, $f(x) = 2$ if $x = 1$: Removable discontinuity at $x = 1$.

• For $g(x) = \frac{x^2 - 2}{x - 2}$ if $x \neq 2$, $g(x) = 1$ if $x = 2$: Jump discontinuity at $x = 2$.

• For $h(x) = \frac{1}{x^3}$: Infinite/Essential discontinuity as $x \to 0$.

Summary Table: Types of Discontinuities

Type	Description	Example
Removable	Limit exists, $f(a)$ not defined or mismatched	$\frac{x^2 - 1}{x - 1}$ at $x = 1$
Jump	Left and right limits differ	Piecewise function with different values at a point
Infinite/Essential	Limit does not exist, function diverges or oscillates	$\frac{1}{x^3}$ at $x = 0$

Additional info:

• Piecewise functions can be continuous if the left and right limits match at the transition point.

• Oscillatory essential discontinuities can occur for functions like $f(x) = \sin(1/x)$ as $x \to 0$.