Infinite Limits and Limits at Infinity

Understanding how functions behave as they grow infinitely large or as the input approaches a critical value is a core concept in calculus. These behaviors define a function's asymptotes.

Infinite Limits (Vertical Asymptotes)

An infinite limit occurs when the y-value of a function $f(x)$ grows without bound (approaching $\infty$ or $-\infty$) as $x$ approaches a specific finite value $c$. This behavior creates a vertical asymptote at $x = c$.

Example: Evaluate $\lim_{x \to 2^+} \frac{1}{x - 2}$.

As $x$ approaches $2$ from the right (values slightly larger than 2, like 2.01), the denominator $x - 2$ becomes a very small positive number. Dividing 1 by a tiny positive number yields a massive positive number.

$\lim_{x \to 2^+} \frac{1}{x - 2} = \infty$

Similarly, approaching from the left ($x \to 2^-$) yields $-\infty$. Therefore, the line $x = 2$ is a vertical asymptote.

Limits at Infinity (Horizontal Asymptotes)

A limit at infinity describes the "end behavior" of a function—what happens to $f(x)$ as $x$ itself gets infinitely large ($x \to \infty$ or $x \to -\infty$). If this limit is a finite number $L$, the function has a horizontal asymptote at $y = L$.

For rational functions (a polynomial divided by a polynomial), you can quickly evaluate the limit at infinity by comparing the highest degree (power) of the numerator and the denominator:

1. Numerator degree < Denominator degree: The limit is $0$. (Horizontal asymptote at $y = 0$).
2. Numerator degree = Denominator degree: The limit is the ratio of the leading coefficients.
3. Numerator degree > Denominator degree: The limit is $\infty$ or $-\infty$ (No horizontal asymptote).

Example: Evaluate $\lim_{x \to \infty} \frac{3x^2 + 1}{2x^2 - x}$.

Here, the highest power of $x$ in both the numerator and denominator is $x^2$. Because the degrees are equal, the limit is simply the ratio of their leading coefficients (3 and 2).

$\lim_{x \to \infty} \frac{3x^2 + 1}{2x^2 - x} = \frac{3}{2}$

Thus, the function levels off, creating a horizontal asymptote at $y = \frac{3}{2}$.