Determine the following limits.


b. $\underset{x\to 3}{\lim }\frac{x-3}{x^4-9x^2}$

Verified step by step guidance

1. Identify the limit expression: \( \lim_{x \to 3} \frac{x-3}{x^4 - 9x^2} \).
2. Notice that direct substitution of \( x = 3 \) results in an indeterminate form \( \frac{0}{0} \).
3. Factor the denominator: \( x^4 - 9x^2 = x^2(x^2 - 9) = x^2(x-3)(x+3) \).
4. Cancel the common factor \( (x-3) \) from the numerator and the denominator.
5. Re-evaluate the limit with the simplified expression: \( \lim_{x \to 3} \frac{1}{x^2(x+3)} \).

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. In this case, we are interested in the limit of a fraction as x approaches 3, which requires evaluating the function's behavior close to that point.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to yield the original polynomial. This technique is essential when evaluating limits, especially when direct substitution leads to indeterminate forms like 0/0. In the given limit, factoring the denominator will help simplify the expression and resolve the limit.

Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to ambiguous results, such as 0/0 or ∞/∞. These forms require further analysis or manipulation to resolve. In this problem, substituting x = 3 directly into the limit results in an indeterminate form, necessitating the use of algebraic techniques like factoring or L'Hôpital's rule to find the actual limit.