1. **Problem Statement:** Find the limit of a function as $x$ approaches infinity for various types of functions. 2. **General Idea:** The limit at infinity describes the behavior of a function as $x$ becomes very large (positively or negatively). 3. **Common Cases and Formulas:** - For rational functions $\lim_{x \to \infty} \frac{P(x)}{Q(x)}$, where $P$ and $Q$ are polynomials, the limit depends on the degrees of $P$ and $Q$: - If degree($P$) < degree($Q$), limit is 0. - If degree($P$) = degree($Q$), limit is ratio of leading coefficients. - If degree($P$) > degree($Q$), limit is $\pm \infty$ depending on signs. - For exponential functions $\lim_{x \to \infty} a^x$ with $a>1$, limit is $\infty$. - For exponential decay $0 1, limit is $\infty$. 7. **Example 4:** $\lim_{x \to \infty} \frac{\log(x)}{x}$ - Logarithm grows slower than any polynomial. - Limit is 0. **Summary:** To find limits at infinity, compare growth rates of numerator and denominator or use known limits of exponential and logarithmic functions. **Final answers:** - Example 1: $\frac{3}{2}$ - Example 2: 0 - Example 3: $\infty$ - Example 4: 0