Determine whether the following statements are true and give an explanation or counterexample.


a. The value of $\underset{x\to 3}{\lim }\frac{x^2-9}{x-3}$ does not exist.

Verified step by step guidance

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

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. For example, the limit of a function can exist even if the function itself does not take a value at that point.

Indeterminate Forms

Indeterminate forms arise in calculus when evaluating limits that do not lead to a clear value, such as 0/0 or ∞/∞. These forms require further analysis, often using algebraic manipulation or L'Hôpital's Rule, to resolve the limit. Recognizing these forms is crucial for determining the existence of limits in complex expressions.

Factoring and Simplifying Expressions

Factoring and simplifying expressions is a technique used to rewrite complex algebraic expressions in a more manageable form. In the context of limits, this often involves canceling common factors to eliminate indeterminate forms. For instance, the expression (x^2 - 9)/(x - 3) can be factored to (x - 3)(x + 3)/(x - 3), allowing for simplification and easier limit evaluation.