Topic 1.15 Notes – Connecting Limits at Infinity and Horizontal Asymptotes

1. Limits at Infinity and End Behavior

When you see

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

you are asking:

As $x$ gets very large (positive or negative), what does $f(x)$ approach?

This is end behavior.

A few possibilities:

• $\lim_{x \to \infty} f(x) = L$
  → The function levels off toward a finite number $L$.
• $\lim_{x \to \infty} f(x) = \infty$ or $-\infty$
  → The function grows or decreases without bound.
• The function keeps bouncing around (like $\sin x$)
  → The limit does not exist (DNE).

Important:
You are not plugging in infinity. Infinity describes behavior, not a number.

Here’s what this looks like visually. In this example, the function approaches a horizontal line as $x \to \infty$ and as $x \to -\infty$.

End behavior of $f(x) = 3 + \frac{2}{x}$

As $x$ gets large in either direction, the graph flattens toward $y = 3$. That flattening is what the limit at infinity captures.

2. Horizontal Asymptotes and What They Mean

A horizontal asymptote (HA) is a line $y = L$ that the function approaches as $x \to \infty$ and/or $x \to -\infty$.

The connection is direct:

• If $\lim_{x \to \infty} f(x) = L$, then $y = L$ is a horizontal asymptote (on the right).
• If $\lim_{x \to -\infty} f(x) = L$, then $y = L$ is a horizontal asymptote (on the left).

A few truths students mix up:

• ✔️ A function can cross a horizontal asymptote.
• ✔️ The left and right limits can be different.
• ✔️ If the limit is not finite, there is no horizontal asymptote in that direction.

On multiple-choice questions, they love giving you a graph and asking for
$\lim_{x \to -\infty} f(x)$.
Look at the left tail only. Don’t guess from the right side.

3. Finding Horizontal Asymptotes for Rational Functions

For a rational function

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

end behavior depends on the degrees of the polynomials.

Case 1: Degree of numerator < degree of denominator

$$\frac{\text{lower power}}{\text{higher power}} \to 0$$

• $\lim_{x \to \pm\infty} f(x) = 0$
• HA is $y = 0$

Example:

$$\lim_{x \to \infty} \frac{5x+1}{x^2+4} = 0$$

The $x^2$ in the denominator dominates.

Case 2: Degrees are equal

Keep only leading terms.

$$\frac{ax^n}{bx^n} \to \frac{a}{b}$$

• HA is ratio of leading coefficients.

Example:

$$\lim_{x \to \infty} \frac{7x^3 - 2}{2x^3 + 9} = \frac{7}{2}$$

Lower terms become insignificant as $x$ grows.

Case 3: Degree of numerator > degree of denominator

The top grows faster.

• Limit is $\pm\infty$
• No horizontal asymptote

Example:

$$\lim_{x \to \infty} \frac{x^3}{4x} = \infty$$

On free-response, justification usually means writing something like
“highest-degree terms dominate as $x \to \infty$.”

4. Comparing Growth Rates of Functions

Not all functions grow at the same speed.

From slowest to fastest:

$$\log x \;<\; \sqrt{x} \;<\; x^n \;<\; a^x \quad (a>1)$$

That order explains many limits instantly.

Examples

$$\lim_{x \to \infty} \frac{x^4}{e^x} = 0$$

Exponential beats polynomial → denominator wins.

$$\lim_{x \to \infty} \frac{3^x}{x^{10}} = \infty$$

Exponential in numerator grows much faster.

$$\lim_{x \to \infty} \frac{\sqrt{x}}{x} = 0$$

Lower power divided by higher power.

On calculator MCQs, recognizing growth rate saves time. No heavy algebra needed.

5. Special Cases and Common Errors

Oscillation

$$\lim_{x \to \infty} \sin x = \text{DNE}$$

The function never settles to one value. No horizontal asymptote.

Squeeze-Type Behavior

$$\lim_{x \to \infty} \frac{\cos x}{x} = 0$$

The oscillation shrinks because it’s divided by something growing without bound.

Mistakes that cost points

• Thinking horizontal asymptotes can’t be crossed.
• Forgetting to check behavior as $x \to -\infty$.
• Ignoring which term has the highest degree.
• Treating infinity like a number you can substitute.