Solved examples: Limits at infinity for Pre-Calculus students

📚 Quick Study Guide

• ♾️ Limits at Infinity: Describe the behavior of a function $f(x)$ as $x$ approaches infinity ($\infty$) or negative infinity $(-\infty$).
• 📈 Horizontal Asymptotes: If $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$, then $y = L$ is a horizontal asymptote of $f(x)$.
• ➗ Rational Functions: For rational functions ($\frac{P(x)}{Q(x)}$), divide both the numerator and denominator by the highest power of $x$ in the denominator.
• 💪 Dominant Terms: Focus on the dominant terms (highest powers of $x$) in the numerator and denominator as $x$ approaches infinity.
• 💡 Rules:
  • If the degree of $P(x)$ < degree of $Q(x)$, then $\lim_{x \to \infty} \frac{P(x)}{Q(x)} = 0$.
  • If the degree of $P(x)$ = degree of $Q(x)$, then $\lim_{x \to \infty} \frac{P(x)}{Q(x)} =$ ratio of leading coefficients.
  • If the degree of $P(x)$ > degree of $Q(x)$, then $\lim_{x \to \infty} \frac{P(x)}{Q(x)} = \infty$ or $-\infty$ (determine the sign).

Practice Quiz

1. What is the value of $\lim_{x \to \infty} \frac{3x^2 + 2x + 1}{x^2 + 5}$?
   1. 0
   2. 3
   3. $\infty$
   4. 1/5
2. What is the value of $\lim_{x \to \infty} \frac{4x + 3}{x^2 + 1}$?
   1. 0
   2. 4
   3. $\infty$
   4. 3
3. What is the value of $\lim_{x \to -\infty} \frac{2x^3 - 1}{5x^3 + x}$?
   1. 0
   2. 2/5
   3. $\infty$
   4. -1/5
4. What is the value of $\lim_{x \to \infty} \frac{x^4 + 1}{x^2 - 2}$?
   1. 0
   2. 1
   3. $\infty$
   4. -1/2
5. What is the value of $\lim_{x \to \infty} \frac{\sqrt{x} + 1}{x - 3}$?
   1. 0
   2. 1
   3. $\infty$
   4. -1/3
6. What is the horizontal asymptote of $f(x) = \frac{5x}{2x + 1}$?
   1. y = 0
   2. y = 5/2
   3. y = 1/2
   4. No horizontal asymptote
7. What is the value of $\lim_{x \to -\infty} \frac{-3x^5 + 2x}{x^5 - 4x^2 + 1}$?
   1. 0
   2. -3
   3. $\infty$
   4. 3