Topic 1.14 Notes – Connecting Infinite Limits and Vertical Asymptotes

Infinite Limits and What They Mean

You already know a limit usually asks, “What value is $f(x)$ approaching as $x$ approaches $a$?”

Now we extend that idea.

If

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

the function does not approach a number. It grows larger and larger without bound as $x$ gets close to $a$.

If

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

the function decreases without bound.

Important details:

• $\infty$ is not a real number.
• The limit does not exist as a finite value, but it still tells us meaningful behavior.
• One-sided limits matter:
  • $\lim_{x \to a^+} f(x)$ → from the right
  • $\lim_{x \to a^-} f(x)$ → from the left

Sometimes one side goes to $+\infty$ and the other to $-\infty$. That difference matters.

This “unbounded behavior” is exactly what creates a vertical asymptote.

Vertical Asymptotes and Their Connection to Infinite Limits

A vertical asymptote is a vertical line $x = a$ where the function’s values grow without bound.

Here’s the connection:

If either

• $\lim_{x \to a^+} f(x) = \pm\infty$, or
• $\lim_{x \to a^-} f(x) = \pm\infty$,

then $x = a$ is a vertical asymptote.

The function:

• Is undefined at $x = a$
• Is not continuous there
• Has an infinite discontinuity

Here are the three common behaviors. In the graphs below, focus on the vertical line $x = 0$.

For $\frac{1}{x}$, the function goes to $-\infty$ on one side and $+\infty$ on the other. For $\frac{1}{x^2}$, it goes to $+\infty$ on both sides. In each case, the graph never actually touches the asymptote. It just grows without bound near it.

On FRQs, you may be asked to justify a vertical asymptote using limit notation. That means writing the correct one-sided limit statements.

How to Find Vertical Asymptotes Algebraically

Most of the time, you’ll see this with rational functions.

Step 1: Find where the function is undefined

Common triggers:

• Denominator equals 0
• Log input equals 0
• Even root of a negative number

Example:

$$f(x) = \frac{2}{x - 5}$$

The denominator is 0 at $x = 5$. That’s your suspect.

Step 2: Evaluate one-sided behavior

Pick values slightly less than 5 and slightly greater than 5.

• If $x = 4.9$, denominator is negative → fraction is large negative.
• If $x = 5.1$, denominator is positive → fraction is large positive.

So:

$$\lim_{x \to 5^-} f(x) = -\infty \quad\text{and}\quad \lim_{x \to 5^+} f(x) = \infty$$

Therefore, $x = 5$ is a vertical asymptote.

On multiple choice, they often test whether you understand the sign on each side. Always check it.

Factoring First Matters

If you can factor and cancel, check that before declaring an asymptote.

Example idea:

$$\frac{x-3}{(x-3)(x+1)}$$

The $x-3$ cancels. That creates a hole, not a vertical asymptote at 3.

After simplifying, if a zero is still in the denominator, then it’s a vertical asymptote.

Students lose easy points here by skipping simplification.

Common Functions with Vertical Asymptotes

Rational functions

Vertical asymptotes occur where the denominator equals 0 after simplification.

Logarithmic functions

$$\lim_{x \to 0^+} \ln(x) = -\infty$$

So $x = 0$ is a vertical asymptote.

Tangent function

Vertical asymptotes occur where cosine equals 0, like $x = \frac{\pi}{2}$.

Any time the output becomes unbounded near a specific $x$-value, you’re looking at infinite limits and vertical asymptotes.

Interpreting From a Graph

If a graph shoots upward near $x = a$, the limit is $+\infty$. If it drops downward, the limit is $-\infty$.

That tells you:

• The function is not continuous
• The limit does not exist as a finite number
• There is an infinite discontinuity

On the AP exam, they love asking you to interpret this from a graph and write the correct limit notation. Pay attention to which side you’re approaching from.