1. Let's start by understanding the problem: you want to know when to factorize and when to take the highest power when evaluating limits. 2. When dealing with limits, especially as $x \to \infty$ or $x \to -\infty$, the behavior of the function is often dominated by the term with the highest power of $x$. 3. **Taking the highest power:** - This is useful when you want to find the limit of a rational function (a ratio of polynomials) as $x$ approaches infinity or negative infinity. - The highest power terms in the numerator and denominator determine the end behavior. - For example, for $\lim_{x \to \infty} \frac{3x^4 + 5x^2}{2x^4 - x + 1}$, the highest power is $x^4$ in both numerator and denominator. - You can factor out $x^4$ from numerator and denominator to simplify. 4. **Factorizing:** - Factorization is useful when you want to simplify expressions, especially to cancel common factors. - This is often used when the limit leads to an indeterminate form like $\frac{0}{0}$. - For example, $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ can be factorized as $\frac{(x-2)(x+2)}{x-2}$, then cancel $x-2$ to get $\lim_{x \to 2} (x+2) = 4$. 5. **Summary:** - Use highest power to analyze limits at infinity. - Use factorization to simplify expressions and resolve indeterminate forms. 6. Remember, the goal is to simplify the expression to evaluate the limit easily.