1.14 Connecting Infinite Limits and Vertical Asymptotes

A vertical asymptote shows up at $x = a$ when a function's value grows without bound toward $+\infty$ or $-\infty$ as $x$ approaches $a$. You confirm this with an infinite limit: if a one-sided or two-sided limit at $x = a$ equals $\pm\infty$, then $x = a$ is a vertical asymptote and the function is discontinuous there. For AP Calculus, check both sides of the asymptote because the signs can differ.

Why This Matters for the AP Calculus Exam

Infinite limits connect three things the AP Calculus exam loves to test together: limits, discontinuities, and graph behavior. When you can read a function's unbounded behavior through limit notation, you can describe and explain what a graph does near a trouble spot instead of just guessing from a picture.

This skill supports the bigger goal of interpreting function behavior using limits involving infinity. You'll use it to identify vertical asymptotes on rational, logarithmic, and trigonometric functions, to classify a discontinuity as an infinite (asymptotic) type, and to match graphical, numerical, and analytic representations of the same behavior. Limits like these also set up later work with end behavior, curve sketching, and analyzing functions.

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Key Takeaways

• An infinite limit means the function grows or drops without bound: $\lim_{x\to a} f(x) = \pm\infty$.
• If a one-sided limit, $\lim_{x\to a^+} f(x)$ or $\lim_{x\to a^-} f(x)$, equals $\pm\infty$, then $x = a$ is a vertical asymptote.
• A vertical asymptote is a discontinuity because the function is unbounded and undefined there.
• For rational functions, vertical asymptotes happen where the denominator is zero and the factor does not cancel.
• Saying a limit "equals $\infty$" describes unbounded behavior; the limit still does not exist as a finite real number.
• Check the left side and the right side separately, since a function can go to $+\infty$ on one side and $-\infty$ on the other.

Discontinuities and Vertical Asymptotes

Discontinuities are points where a function is undefined or suddenly changes behavior. Many calculus theorems only apply when a function is continuous, so being able to spot discontinuities matters.

A vertical asymptote is a vertical line that a function approaches but never crosses. Vertical asymptotes are a type of discontinuity because they mark an $x$-value where the function's behavior is unbounded.

Here is what unbounded means in practice: as $x$ gets really close to that $x$-value, the $y$-value rapidly increases or decreases toward infinity, producing the steep vertical-looking part of the graph. The function cannot be evaluated at that exact $x$-value, so it is a discontinuity.

Infinite Limits

Limits that evaluate to positive or negative infinity are infinite limits. A function increases without bound for positive infinity and decreases without bound for negative infinity.

Connecting Infinite Limits to Vertical Asymptotes

If any of these are true, then $x = a$ is a vertical asymptote:

1. $\lim_{x\to a} f(x) = \pm\infty$
2. $\lim_{x\to a^+} f(x) = \pm\infty$
3. $\lim_{x\to a^-} f(x) = \pm\infty$

In words: if a function $f$ heads toward infinity as $x$ approaches a value $a$, there is a vertical asymptote at $x = a$. The function might blow up from the left ($-$), the right ($+$), or both sides.

How to Use This on the AP Calculus Exam

MCQ

• Spot vertical asymptotes fast: in a rational function, find where the denominator is zero and the factor does not cancel. If it cancels, you have a hole (removable discontinuity), not an asymptote.
• Use sign analysis to decide $+\infty$ versus $-\infty$. Plug in a value just to the left and just to the right of $x = a$ and track the sign of the result.
• Recognize asymptotic behavior across representations. A table showing values like 100, 10000, 1000000 as $x$ approaches $a$ signals an infinite limit even without a graph.

Free Response

• When you write that $x = a$ is a vertical asymptote, back it up with limit notation, such as $\lim_{x\to a^+} f(x) = \infty$. Stating the limit is clearer and stronger than just naming the line.
• Keep your one-sided notation precise. Writing $a^+$ versus $a^-$ tells the reader exactly which side you analyzed.

Common Trap

• "The limit equals infinity" is a description of unbounded behavior, not a finite answer. If a question asks whether the limit exists as a real number, the answer is no.

Worked Examples

Example 1: Confirming a Vertical Asymptote

Using limits, show that $x = -3$ is a vertical asymptote for $f(x) = \frac{1}{x+3}$.

If $x = -3$ is a vertical asymptote, then the limit as $x$ approaches $-3$ must be positive or negative infinity.

Check the right side: $\lim_{x\to -3^+} \frac{1}{x+3} = \infty$

Substituting $-3$ makes the denominator 0, so 1 is divided by something approaching 0. Approaching from the right means $x$ is just slightly greater than $-3$ (think $-2.99999$), so $x + 3$ is a tiny positive number. Dividing 1 by a tiny positive number gives a huge positive value, so the function grows toward $+\infty$.

Check the left side: $\lim_{x\to -3^-} \frac{1}{x+3} = -\infty$

Approaching from the left means $x$ is just slightly less than $-3$ (think $-3.00001$), so $x + 3$ is a tiny negative number. Dividing 1 by a tiny negative number gives a huge negative value, so the function drops toward $-\infty$.

Either one-sided limit being infinite is enough to confirm a vertical asymptote at $x = -3$.

Example 2: A Logarithmic Vertical Asymptote

Find the vertical asymptote for $f(x) = \ln(x)$.

From the shape of the natural log graph, the vertical asymptote is at $x = 0$. Confirm it with an infinite limit:

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

$\ln(0)$ does not exist because no value of $n$ makes $e^n = 0$. As $x$ gets smaller and approaches 0 from the right, $\ln(x)$ becomes more and more negative. Since $e \approx 2.718$, getting $e^n$ close to 0 requires $n$ to be very negative, which is why $\ln(x) \to -\infty$.

Note that $\ln(x)$ is only defined for $x > 0$, so you only check the right-hand limit here.

Common Misconceptions

• A zero in the denominator always means a vertical asymptote. Not true. If the same factor cancels with the numerator, you get a hole (removable discontinuity), not an asymptote. Only a non-canceling zero in the denominator gives a vertical asymptote.
• An infinite limit means the limit exists. Saying $\lim_{x\to a} f(x) = \infty$ describes unbounded behavior. The limit does not exist as a finite real number.
• The function approaches the same infinity on both sides. A function can go to $+\infty$ from one side and $-\infty$ from the other, like $\frac{1}{x+3}$. Always check each side.
• A function crosses its vertical asymptote. It approaches the line but never touches or crosses it at that $x$-value, which is exactly why the point is a discontinuity.
• Vertical asymptotes only come from fractions. Logarithmic functions like $\ln(x)$ also have vertical asymptotes, and so do functions like $\tan x$ at odd multiples of $\frac{\pi}{2}$.

Related AP Calculus Guides

• 1.2 Defining Limits and Using Limit Notation
• 1.1 Introducing Calculus: Can Change Occur at An Instant?
• 1.6 Determining Limits Using Algebraic Manipulation
• 1.3 Estimating Limit Values from Graphs
• 1.5 Determining Limits Using Algebraic Properties of Limits
• 1.8 Determining Limits Using the Squeeze Theorem