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

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

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Identify the limit expression: $ \lim_{x \to 3} \frac{x^2 - 9}{x - 3} $.

Notice that direct substitution of $ x = 3 $ results in a $ \frac{0}{0} $ indeterminate form.

Factor the numerator: $ x^2 - 9 = (x - 3)(x + 3) $.

Rewrite the limit expression: $ \lim_{x \to 3} \frac{(x - 3)(x + 3)}{x - 3} $.

Cancel the common factor $ x - 3 $ in the numerator and denominator, then evaluate the limit of the simplified expression.

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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.

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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.

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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.

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