Fall 2025: Chapter 2-6 Notes - Limits at Infinity and Asymptotes

Welcome to Professor Baker's Math Class! This section focuses on understanding limits at infinity and how they relate to the asymptotes of a function. Mastering these concepts will allow you to analyze the behavior of functions as $x$ approaches infinity or negative infinity. Let's dive in!

Topics Covered:

  • Limits at infinity and graphs
  • Limits at infinity and rational functions
  • Limits at infinity and functions involving a sum or quotient with a radical
  • Evaluating limits to find the vertical and horizontal asymptotes of a function

Horizontal Asymptotes

A crucial concept when dealing with limits at infinity is the horizontal asymptote. Let's define it formally:

Definition: The line $y = L$ is called a horizontal asymptote of the curve $y = f(x)$ if either:

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

In simpler terms, as $x$ gets very large (either positively or negatively), the function $f(x)$ approaches the value $L$.

Limits at Infinity and Rational Functions

When evaluating limits at infinity for rational functions, we primarily focus on the degrees of the polynomials in the numerator and the denominator. Here are some key observations:

  • If the degree of the numerator (N) is greater than the degree of the denominator (D), then there is no horizontal asymptote. However, we must examine if $ \lim_{x \to \infty} f(x) = \infty $ or $ -\infty $
  • If the degree of the numerator (N) is less than the degree of the denominator (D), then $y = 0$ is the horizontal asymptote, which means $ \lim_{x \to \infty} f(x) = 0 $.
  • If the degree of the numerator (N) is equal to the degree of the denominator (D), then $y =$ (leading coefficient of numerator) / (leading coefficient of denominator) is the horizontal asymptote.

Example:

Let's find the limit of the following rational function as $x$ approaches negative infinity:

$$ \lim_{x \to -\infty} \frac{6 - 5x^2}{2x^3 - 7} $$

Divide both the numerator and the denominator by $x^3$:

$$ \lim_{x \to -\infty} \frac{\frac{6}{x^3} - \frac{5}{x}}{2 - \frac{7}{x^3}} $$

Since $ \lim_{x \to -\infty} \frac{1}{x^n} = 0 $ for $n > 0$, we have:

$$ \frac{0 - 0}{2 - 0} = 0 $$

Therefore, $y = 0$ is the horizontal asymptote.

Limits Involving Radicals

When dealing with limits at infinity involving radicals, a common technique is to multiply by the conjugate or to factor out the dominant term. Remember that when taking a variable inside a square root, you may need to consider absolute values if $x$ is approaching negative infinity.

Keep practicing, and you'll master these concepts in no time! Good luck with your studies!