Calculus I

Limits at Infinity and Asymptotes

Imron Rosyadi

Limits at Infinity

Understanding Limits at Infinity

So far we’ve looked at limits as \(x \to a^+\), \(x \to a^-\) and \(x \to a\). Now we explore what happens as \(x\) moves indefinitely, either positively or negatively.

As \(x \to \infty\)

This describes \(f(x)\)’s behavior as \(x\) grows without bound in the positive direction.

Think of \(x\) as 100, then 1,000, then 1,000,000, and so on.

If \(\displaystyle{\lim_{x \to \infty}} f(x)=L\), it means \(f(x)\) gets arbitrarily close to \(L\) as \(x\) becomes very large.

As \(x \to -\infty\)

This is similar, but in the negative direction.

Imagine \(x\) as -100, then -1,000, then -1,000,000.

If \(\displaystyle{\lim_{x \to -\infty}} f(x)=L\), then \(f(x)\) approaches \(L\) as \(x\) becomes very large negatively.

Tip

End Behavior. Limits at infinity help us understand the “end behavior” of a function’s graph, seeing where it settles as \(x\) goes far to the left or right.

Horizontal Asymptotes

When a function approaches a specific value as \(x \to \pm \infty\), that value defines a horizontal asymptote.

Definition: The line \(y=L\) is called a horizontal asymptote for \(y=f(x)\) if and only if \[ \lim_{x\to\infty}f(x)=L, \quad \text{ or }\quad \lim_{x\to-\infty}f(x)=L \]

Important

Can a function have more than one horizontal asymptote? Yes, a function can have at most two distinct horizontal asymptotes (one as \(x \to \infty\) and one as \(x \to -\infty\)).

The graph on the left has \(y=\pi/2\) and \(y=-\pi/2\) as horizontal asymptotes.

The graph on the right has horizontal asymptotes \(y=\pm 4\).

Explain that horizontal asymptotes describe the long-term behavior of a function. Unlike vertical asymptotes, a function can cross its horizontal asymptote multiple times; it’s only constrained as \(x\) approaches infinity or negative infinity. For example, damped oscillations cross the x-axis (a horizontal asymptote) many times.

Limits of Rational Functions

A rational function is defined as \(f(x) = \dfrac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials.

Evaluating limits of rational functions as \(x \to \pm \infty\) is a fundamental concept.

It reveals insights into the high school “rule” of comparing the degrees of the numerator and denominator to find horizontal asymptotes.

To evaluate \(\displaystyle \lim_{x\to\infty} \frac{P(x)}{Q(x)}\):

Divide both the numerator and the denominator by the highest power of \(x\) in the denominator.

Then simplify and apply the property \(\displaystyle \lim_{x\to\infty} \frac{1}{x^n} = 0\) for \(n > 0\).

Note

Even with the “rule,” understanding the process of evaluating these limits is crucial for deeper comprehension.

Limits of Rational Functions

Numerical Exploration

Let’s observe the behavior of a rational function as \(x\) becomes very large.

Discuss how the output from the Pyodide block shows the function’s value approaching 3, which is the ratio of the leading coefficients. This visually confirms the rule for rational functions where degrees are equal. Emphasize that for students, the algebraic method is the primary one, but numerical checks can build intuition.

Functions Growth Rates

To compute limits like \(\displaystyle{\lim_{x \to \infty} \frac{f(x)}{g(x)}}\), we need to compare the growth rates of \(f(x)\) and \(g(x)\). This also involves identifying the fastest-growing part within each function.

Tip

Dominant Terms. When \(x\) approaches infinity, terms with higher powers or faster-growing functions (like exponentials) “dominate” the behavior. The other terms become negligible.

Hierarchy of Growth Rates

Understanding which functions grow faster is key to evaluating complex limits.

graph LR
    A["Slowest: log(x)"] --> B["Polynomials: x^n (n>0)"];
    B --> C["Exponentials: a^x (a>1)"];
    C --> D["Factorials: x!"];
    D --> E["Fastest: x^x"];
    style A fill:#f9f,stroke:#333,stroke-width:2px;
    style B fill:#bbf,stroke:#333,stroke-width:2px;
    style C fill:#bfb,stroke:#333,stroke-width:2px;
    style D fill:#fbb,stroke:#333,stroke-width:2px;
    style E fill:#ff9,stroke:#333,stroke-width:2px;

Elaborate on each category in the growth hierarchy. For example, explain why \(e^x\) grows faster than \(x^{100}\), or why \(\ln(x)\) grows slower than \(x^{0.001}\). Provide simple examples for each type. Mention L’Hôpital’s Rule as a tool for comparing growth rates in indeterminate forms, but note that it’s typically introduced later.

Example: Evaluating a Limit at Negative Infinity

Evaluate \(\displaystyle \lim_{x\to-\infty}\frac{x+3}{\sqrt{9x^2-5x}}\).

Solution Steps

1. Identify Dominant Terms:
   • As \(x\to-\infty\), the dominant term in the numerator is \(x\).
   • In the denominator, \(x^2\) is the dominant term within the square root. So, \(\sqrt{9x^2-5x} \sim \sqrt{9x^2}\).
2. Simplify the Denominator:
   • \(\sqrt{9x^2} = 3\sqrt{x^2} = 3|x|\).
3. Address Negative Infinity:
   • Since \(x\to-\infty\), \(x\) is negative. Therefore, \(|x| = -x\).
   • So, \(\sqrt{9x^2-5x} \sim 3(-x) = -3x\).
4. Form the Simplified Ratio:
   • The expression becomes approximately \(\frac{x}{-3x}\).
5. Evaluate the Limit:
   • \(\displaystyle \lim_{x\to-\infty}\frac{x}{-3x} = \lim_{x\to-\infty}\left(-\frac13\right) = -\frac13\).

Thus, the value of the limit is \(-\frac13\). This means \(y=-1/3\) is a horizontal asymptote.

Tip

Remember the absolute value! When dealing with \(\sqrt{x^2}\), it always simplifies to \(|x|\). The direction of infinity (positive or negative) dictates whether \(|x|\) is \(x\) or \(-x\). This is a common pitfall.

Walk through each step clearly. Emphasize the reasoning for dropping lower-order terms and especially the correct handling of \(\sqrt{x^2}\) as \(|x|\) and then as \(-x\) for negative infinity. This is a critical point that students often miss.

Vertical Asymptotes (Redux)

What happens if a function \(f(x)\) grows (or decreases) without bound as \(x\) approaches some finite value?

• If \(\displaystyle{\lim_{x \to a^+} f(x) = \infty}\), then \(f(x)\) is large and positive whenever \(x\) is slightly greater than \(a\).
• This indicates the graph \(y=f(x)\) gets very close to the vertical line \(x=a\). We call this a vertical asymptote.
• The same occurs if \(\displaystyle{\lim_{x \to a^+} f(x) = -\infty}\), or if \(\displaystyle{\lim_{x \to a^-} f(x) = \pm \infty}\). In all these cases, \(x=a\) is a vertical asymptote.

Warning

Key Distinction: Vertical asymptotes occur at finite \(x\)-values where the function itself goes to infinity. Horizontal asymptotes occur when \(x\) goes to infinity (or negative infinity), and the function approaches a finite value.

We have seen that \(f(x)=\frac{1}{x}\) has a vertical asymptote. Where is it?

Since \(\displaystyle{\lim_{x \to 0^+} \frac{1}{x} = \infty}\) and \(\displaystyle{\lim_{x \to 0^-} \frac{1}{x} = -\infty}\), there is a vertical asymptote at \(x=0\).

Reinforce the idea that vertical asymptotes are caused by division by zero in rational functions or other singularities, where the function values “blow up.” Contrast this clearly with horizontal asymptotes. Use \(f(x)=1/x\) as a canonical example, sketching its behavior around \(x=0\).

Summary of Asymptotes

A concise overview

1. Limit as \(x \to \infty\) (or \(x \to -\infty\)) describes \(f(x)\)’s behavior as \(x\) increases (or decreases) without bound.
2. The limit laws apply to limits as \(x \to \pm \infty\), just as they apply to limits as \(x \to a\) or \(x \to a^+\) or \(x \to a^-\).
3. An asymptote is a line that the graph \(y=f(x)\) approaches. This line can be vertical, horizontal, or diagonal (oblique - not covered here). A curve can be an asymptote too.
4. A horizontal asymptote \(y=L\) arises if \(\displaystyle{\lim_{x \to \infty} f(x) = L}\) or \(\displaystyle{\lim_{x \to -\infty} f(x) = L}\). A curve may cross its horizontal asymptote.
5. A vertical asymptote \(x=a\) happens if \(\displaystyle{\lim_{x \to a^+} f(x) = \pm \infty}\) or \(\displaystyle{\lim_{x \to a^-} f(x) = \pm \infty}\). A curve cannot cross a vertical asymptote as it is undefined there.

Important

Always state asymptotes as equations of lines (e.g., \(y=3\), not just “3”).

Review each point, perhaps adding a quick mental example for each. Emphasize the difference in “crossing behavior” between horizontal and vertical asymptotes. Briefly mention oblique asymptotes for completeness, stating they occur when the degree of the numerator is exactly one higher than the denominator, but that’s a topic for another day.

Identify the Asymptotes!

DO: You should recognize all of the asymptotes of the following graphs (if they have any). You should know the functions for all but the 2nd and 3rd graphs without looking at the labels.

graph1

graph5

graph9

graph13

graph2

graph6

graph10

graph14