Topic 1.15 Notes – Connecting Limits at Infinity and Horizontal Asymptotes

Limits at Infinity and End Behavior

A limit at infinity asks:

$$\lim_{x \to \infty} f(x) \quad \text{or} \quad \lim_{x \to -\infty} f(x)$$

This describes the function’s end behavior - what the graph does far right or far left.

Three possibilities:

• The limit is a finite number $L$ → the function levels off toward $L$.
• The limit is $\pm\infty$ → the function grows without bound.
• The function never settles (keeps oscillating) → the limit does not exist (DNE).

Important: this is not about behavior near a vertical asymptote or a specific x-value. It’s about long-term trends.

Horizontal Asymptotes

If

$$\lim_{x \to \infty} f(x) = L$$

then the line

$$y = L$$

is a horizontal asymptote (HA) on the right side.

Same idea for $x \to -\infty$.

Here’s what that looks like visually. In this example, the function approaches $y = 2$ as $x \to \pm\infty$, so $y = 2$ is the horizontal asymptote.

Horizontal asymptote $y = 2$

Key ideas students mix up:

• A function can cross a horizontal asymptote.
• You can have two different horizontal asymptotes (different left and right limits).
• Some functions have no horizontal asymptote.

On FRQs, if you’re asked to interpret $\lim_{x\to\infty} f(x)=3$, you should state that the function approaches 3 as x increases without bound. That language matters.

Rational Functions and Degree Rules

For

$$f(x)=\frac{p(x)}{q(x)}$$

where $p$ and $q$ are polynomials, end behavior depends on degrees.

Case 1: Degree of $p$ < Degree of $q$

$$\lim_{x \to \pm\infty} f(x) = 0$$

The denominator grows faster, so the fraction shrinks toward 0.
Horizontal asymptote: $y=0$.

Case 2: Degrees Are Equal

$$\lim_{x \to \pm\infty} f(x) = \frac{\text{leading coefficient of } p}{\text{leading coefficient of } q}$$

Only the highest-power terms matter far out.

Example:

$$\lim_{x\to\infty} \frac{5x^3 - 2x}{-2x^3 + 7}$$

Behaves like

$$\frac{5x^3}{-2x^3} = -\frac{5}{2}$$

So HA is $y=-\frac{5}{2}$.

Case 3: Degree of $p$ > Degree of $q$

The function grows without bound.

$$\lim_{x \to \pm\infty} f(x) = \pm\infty$$

No horizontal asymptote. (There could be a slant asymptote later in the course.)

On multiple choice, they often hide this inside an $\frac{\infty}{\infty}$ form. Compare degrees immediately. Don’t plug in giant numbers.

Comparing Growth Rates

When functions aren’t just polynomials, think about relative growth.

From slowest to fastest:

$$\log x \;<\; \sqrt{x} \;<\; x^n \;<\; a^x$$

Logs < roots < polynomials < exponentials.

The graph below shows all four on the same axes so you can see how they separate as $x$ increases.

Relative growth of $\ln x$, $\sqrt{x}$, $x^2$, and $2^x$

Notice how $2^x$ eventually shoots upward, the polynomial rises steadily, the square root increases slowly, and the logarithm barely climbs. As $x \to \infty$, those gaps only get bigger.

How this helps:

• $\frac{x^4}{e^x} \to 0$ (exponential in denominator wins)
• $\frac{3^x}{x^{10}} \to \infty$ (exponential in numerator wins)

The faster-growing function dominates the limit.

This shows up a lot in calculator-active MCQs where you’re comparing expressions quickly.

Oscillation and the Squeeze Idea

Not every function settles.

$$\lim_{x\to\infty} \sin x$$

does not exist because it keeps bouncing between -1 and 1.

But if an oscillating function is divided by something growing:

$$\lim_{x\to\infty} \frac{\cos x}{x}$$

The numerator stays between -1 and 1, while the denominator grows. The whole fraction gets squeezed to 0.

That “bounded over unbounded” pattern is common.