Problem 45 - Thomas Calculus

Question

Prove the limit statements in Exercises \(37-50\) $$ \lim _{x \rightarrow-3} \frac{x^{2}-9}{x+3}=-6 $$

Step-by-Step Solution

Answer

The limit is \(-6\) by simplifying and evaluating the expression.

Step 1: Analyze the Expression

First, we look at the expression \( \frac{x^2 - 9}{x+3} \). Notice that the numerator, \( x^2 - 9 \), can be factored as \( (x - 3)(x + 3) \). Thus, the expression can be rewritten as \( \frac{(x - 3)(x + 3)}{x + 3} \).

Step 2: Simplify the Expression

Since \( x + 3 \) appears in both the numerator and the denominator, we can cancel \( x + 3 \) from both parts, simplifying the expression to \( x - 3 \) for \( x eq -3 \).

Step 3: Evaluate the Limit

After simplifying, the expression is \( x - 3 \). Now we find the limit as \( x \) approaches \(-3\). Substitute \(-3\) into the simplified expression: \( -3 - 3 = -6 \).

Step 4: State the Conclusion

Based on the simplification and substitution, we find that: \( \lim _{x \rightarrow-3} \frac{x^{2}-9}{x+3} = -6 \). The original limit statement is thus proved.

Key Concepts

Limit Evaluation

Understanding limits is a fundamental concept in calculus and plays a crucial role in evaluating the behavior of functions. A limit essentially helps us understand what value a function approaches as its input approaches a certain point. In this exercise, we aim to prove that \[\lim_{x \rightarrow -3} \frac{x^{2}-9}{x+3} = -6.\]
This involves determining the function's behavior as \(x\) gets infinitely close to \(-3\). A direct substitution in the original equation would result in an indeterminate form \(\frac{0}{0}\). Hence, our task includes performing algebraic manipulations to simplify the expression to a determinate form. Ultimately, the goal is to accurately calculate what the output value of the function converges to as \(x\) approaches \(-3\).

Algebraic Simplification

Algebraic simplification is a powerful tool in the evaluation of limits. When faced with complex expressions, simplifying them can make the evaluation much more straightforward. In the given problem, the expression\[\frac{x^{2}-9}{x+3}\]is initially complicated due to its zero-over-zero indeterminate form when \(x = -3\).
To simplify, we take advantage of factoring. By recognizing that the numerator is a difference of squares, \(x^{2} - 9\) can be rewritten as \((x - 3)(x + 3)\). This reveals that part of the numerator matches the denominator, allowing us to cancel \(x + 3\) from both parts, given our restriction that \(x eq -3\). This simplification process is crucial as it transforms the expression into \(x - 3\), which is much easier to handle for further limit evaluation.

Factoring Polynomials

Factoring polynomials is a central technique in algebra, especially when working with quadratic expressions. The expression \(x^2 - 9\) is a classic example of a "difference of squares" which factors into two binomials: \((x-3)(x+3)\). This process is crucial because:

• It uncovers underlying structures within the expression, allowing for cancellation and simplification.
• It transforms the original expression into a more manageable form.
• It permits a clearer evaluation of limits, focusing only on behaviors that are not indeterminate.

Knowing how to factor effectively assists in removing potential barriers in the evaluation process. Once factored and simplified, calculating limits becomes more focused and less prone to algebraic errors.

Calculus Proofs

A core aspect of calculus is proving the correctness of results, such as limit statements. These proofs require logical thinking and a solid grasp of algebraic manipulation. Beginning with an understanding of limits and simplification, the goal is to show step by step how the manipulation conforms to the rules of calculus.
In this proof, after simplifying the original expression, we substitute \(x = -3\) into the simplified function \(x - 3\), yielding:\[-3 - 3 = -6.\]
The formal statement of proof is achieved by demonstrating that the preparation of the original statement, its transformation, and its evaluation consistently follow mathematical principles. Therefore:\[\lim_{x\rightarrow-3}\frac{x^{2}-9}{x+3} = -6\]is correctly shown by applying the appropriate calculus and algebraic techniques, thus confirming the exercise's solution with complete confidence.