Infinite limits, limits at infinity and asymptotes

Introduction to the Far Reaches of Calculus

Welcome to one of the most visually intuitive parts of AP Calculus BC! In our previous chapters, we looked at what happens to a function as \(x\) approaches a specific, finite number. Now, we are going to push the boundaries. We will explore what happens when a function "blows up" to infinity or when we look at where a function is heading as \(x\) travels forever to the right or left.

Understanding infinite limits and limits at infinity is the key to describing the "end behavior" of functions and identifying those invisible boundary lines we call asymptotes. Don't worry if the idea of "infinity" feels a bit overwhelming at first—in calculus, we treat it more like a journey than a destination!

Section 1: Infinite Limits and Vertical Asymptotes

Sometimes, as \(x\) gets closer and closer to a specific value \(c\), the \(y\)-values of the function grow without bound. Imagine a rocket taking off; it just keeps going up and up. This is what we call an infinite limit.

What is an Infinite Limit?

We say that \(\lim_{x \to c} f(x) = \infty\) if the values of \(f(x)\) increase without bound as \(x\) approaches \(c\). Similarly, if the values decrease without bound (becoming very large negative numbers), we say \(\lim_{x \to c} f(x) = -\infty\).

Note: Technically, because infinity is not a real number, if a limit is infinite, the limit does not exist (DNE). However, on the AP Exam, you should describe the behavior as \(\infty\) or \(-\infty\) to be as specific as possible.

Connecting Limits to Vertical Asymptotes

This behavior creates a Vertical Asymptote (VA).
Definition: The line \(x = c\) is a vertical asymptote of the graph of \(f\) if the limit from the left or the right (or both) is infinite.
Specifically:
\(\lim_{x \to c^+} f(x) = \pm\infty\) OR \(\lim_{x \to c^-} f(x) = \pm\infty\)

How to Find Vertical Asymptotes Analytically

For most rational functions (fractions), a vertical asymptote occurs at values of \(x\) that make the denominator zero but do not make the numerator zero.

Example: Consider \(f(x) = \frac{3}{x - 2}\).
As \(x\) approaches \(2\) from the right (\(x \to 2^+\)), the denominator is a tiny positive number. \(3\) divided by a tiny positive number is a huge positive number.
So, \(\lim_{x \to 2^+} \frac{3}{x - 2} = \infty\). This confirms a vertical asymptote at \(x = 2\).

Quick Tip: If both the numerator and denominator are zero at \(x = c\), you likely have a "hole" (removable discontinuity) rather than a vertical asymptote. Always simplify the fraction first! (Cross-reference: Topic 1.13 covers removing discontinuities).

Key Takeaway: Infinite limits tell us how a function fails to exist at a vertical asymptote. If the limit is infinite, there is a VA at that \(x\)-value.

Section 2: Limits at Infinity and Horizontal Asymptotes

Now, let’s flip our thinking. Instead of \(y\) going to infinity, what if \(x\) goes to infinity? This is called a limit at infinity. We are looking at the "far-right" or "far-left" of the graph.

What is a Limit at Infinity?

We are interested in the value \(L\) that a function approaches as \(x\) becomes very large:
\(\lim_{x \to \infty} f(x) = L\) or \(\lim_{x \to -\infty} f(x) = L\)

Connecting Limits to Horizontal Asymptotes

If the limit as \(x \to \infty\) or \(x \to -\infty\) is a finite number \(L\), then the line \(y = L\) is a Horizontal Asymptote (HA).

Did you know? Unlike vertical asymptotes, which a graph can never touch, a graph can cross its horizontal asymptote many times. The HA only describes what happens at the very ends of the graph.

Finding Horizontal Asymptotes: The "Degree Test"

When dealing with rational functions (fractions of polynomials), you can quickly find the HA by comparing the highest power (degree) of the numerator and denominator:

• Bottom Heavy: If the degree of the denominator is larger, the limit is \(0\). The HA is \(y = 0\).
  Example: \(\lim_{x \to \infty} \frac{x + 5}{x^2 - 1} = 0\)
• Top Heavy: If the degree of the numerator is larger, the limit is \(\infty\) or \(-\infty\). There is no HA.
  Example: \(\lim_{x \to \infty} \frac{x^3}{x^2 + 1} = \infty\)
• Equal Degrees: If the degrees are the same, the limit is the ratio of the leading coefficients.
  Example: \(\lim_{x \to \infty} \frac{5x^2 + 1}{2x^2 - 3x} = \frac{5}{2}\). The HA is \(y = \frac{5}{2}\).

The "Big Dog" Analogy

When \(x\) is a trillion, \(x^2\) is so much bigger than \(x\) that the smaller terms don't really matter. To find the limit at infinity, ignore everything except the term with the highest power in the top and the term with the highest power in the bottom. These are the "Big Dogs." They dictate where the function goes!

Key Takeaway: Limits at infinity (\(x \to \infty\)) determine the horizontal asymptotes (\(y = L\)) of a function.

Section 3: Common Pitfalls and Exam Tips

Don't Mix Them Up!

Students often confuse these two concepts because they both involve the word "infinity."
Vertical Asymptotes: \(x \to c\), Answer is \(\pm\infty\). (Look for \(x = \#\))
Horizontal Asymptotes: \(x \to \pm\infty\), Answer is \(L\). (Look for \(y = \#\))

Transcendental Functions

Be careful with functions like \(e^x\) and \(\arctan(x)\). They have different limits at positive and negative infinity!
\(\lim_{x \to \infty} e^x = \infty\), but \(\lim_{x \to -\infty} e^x = 0\).
This means \(e^x\) has a horizontal asymptote at \(y = 0\) on the left side only.

AP Exam Practice and Notation

• Notation Matters: Always write the full limit notation \(\lim_{x \to \infty} f(x) = L\). Simply writing "\(f(x) \to L\)" is often not enough for full credit on Free-Response Questions (FRQs).
• Justification: If asked to justify a vertical asymptote at \(x = c\), you must show the limit calculation: "There is a VA at \(x = c\) because \(\lim_{x \to c^+} f(x) = \infty\)."
• Calculators: On calculator-active sections, you can "estimate" a limit at infinity by plugging in a very large number (like \(1,000,000\)) and seeing what value the function approaches.

Summary Checklist

1. Vertical Asymptotes: Occur when the \(y\)-value goes to \(\infty\) or \(-\infty\) as \(x\) approaches a constant. Check the zeros of the denominator!

2. Horizontal Asymptotes: Occur when the \(y\)-value approaches a constant as \(x\) goes to \(\infty\) or \(-\infty\). Check the ratio of the leading terms!

3. Continuity: Remember that asymptotes represent a type of non-removable discontinuity. A function cannot be continuous at an \(x\)-value where an asymptote exists.

Keep practicing! Infinity might seem big, but with these tools, you can handle it with ease.