Welcome back to Professor Baker's Math Class! In this session (Spring 2025, Chapter 2-6), we shift our focus from limits at a specific point to Limits at Infinity. We are asking the question: what happens to the $y$-values of a function as $x$ gets infinitely large ($x \to \infty$) or infinitely small ($x o -\infty$)?
This concept is the mathematical foundation for finding Horizontal Asymptotes. Let's dive into the key techniques from our class notes.
1. The Fundamental Theorem
The most important tool in our toolkit for this chapter is the behavior of the reciprocal function. As the denominator gets huge, the fraction gets tiny.
Theorem: If $r > 0$ is a rational number, then:
$$ \lim_{x \to \infty} \frac{1}{x^r} = 0 $$We use this property to simplify complex rational expressions by dividing every term by the highest power of $x$ in the denominator.
2. Strategies for Rational Functions
When dealing with a fraction of polynomials (rational functions), the limit depends heavily on the degrees (highest powers) of the numerator and denominator:
- Equal Degrees: The limit is the ratio of the leading coefficients.
Example from notes: $ \lim_{x \to \infty} \frac{3x^2 - x - 2}{5x^2 + 4x + 1} = \frac{3}{5} $. - Bottom Heavy (Degree of Denom > Num): The limit is $0$ (the horizontal asymptote is $y=0$).
- Top Heavy (Degree of Num > Denom): The limit goes to $\infty$ or $-\infty$ (no horizontal asymptote).
3. Handling Indeterminate Forms
Sometimes direct substitution leads to confusing results like $\infty - \infty$. This is called an indeterminate form. It doesn't mean the limit is zero; it means we need to do more work!
One powerful technique shown in the notes is using the conjugate. For example, to solve $\lim_{x \to \infty} (\sqrt{x^2+1} - x)$, we multiply the numerator and denominator by the conjugate $(\sqrt{x^2+1} + x)$. This clears the radical and reveals the true limit, which in this case approaches $0$.
4. The Tricky Case of Negative Infinity
When taking limits as $x \to -\infty$ involving radicals, be careful! Recall that $\sqrt{x^2} = |x|$.
- If $x$ is positive ($x \to \infty$), then $\sqrt{x^2} = x$.
- If $x$ is negative ($x \to -\infty$), then $\sqrt{x^2} = -x$.
This subtle sign change can flip your final answer, so always pay attention to the direction of the limit!
5. Real-World Application
Why do we care about limits at infinity? They tell us about the steady-state of a system. In our brine tank example, we looked at salt concentration over time:
$$ C(t) = \frac{30t}{200 + t} $$By taking the limit as $t \to \infty$, we found that the concentration stabilizes at 30 g/L. This models what happens in the "long run," a crucial concept in engineering and science.
Keep practicing these algebraic manipulations. Whether it's factoring out an $x^2$, using log laws to combine natural logs, or multiplying by a conjugate, the goal is always to get the expression into a form where we can use that fundamental theorem ($1/x \to 0$). Good luck studying!