Problem 12 - Calculus: Early Transcendentals

Question

Evaluate the limit, if it exists. \( \displaystyle \lim_{x \to -3}\frac{x^2 + 3x}{x^2 - x - 12} \)

Step-by-Step Solution

Answer

The limit is \(\frac{3}{7}\).

Step 1: Factor the Expressions

First, let's factor both the numerator and the denominator. The numerator is \(x^2 + 3x\), and this can be factored as \(x(x + 3)\). The denominator is \(x^2 - x - 12\), which factors into \((x - 4)(x + 3)\).

Step 2: Cancel Common Factors

Notice that both the numerator and the denominator have a common factor \((x + 3)\). Therefore, we can cancel this factor out, simplifying the expression to \(\frac{x}{x - 4}\).

Step 3: Evaluate the Limit

Now, evaluate the limit of the simplified expression \(\lim_{x \to -3} \frac{x}{x - 4}\). Substitute \(-3\) into the expression: \(\frac{-3}{-3 - 4} = \frac{-3}{-7} = \frac{3}{7}\).

Step 4: Conclude the Evaluation

Since the fractional value obtained does not lead to any indeterminate form or redefine the limits, the limit exists and is \(\frac{3}{7}\).

Key Concepts

Factoring Polynomials

Factoring polynomials is an important first step in evaluating limits involving rational expressions. A polynomial is factored when it is expressed as a product of its constituent polynomials of lower degrees. This simplifies the expressions, making further calculations manageable. In the provided exercise, we begin by factoring the numerator and the denominator.

• **Numerator:** The expression is given as \(x^2 + 3x\). This can be factored by pulling out the common factor \(x\), resulting in \(x(x + 3)\).
• **Denominator:** The expression \(x^2 - x - 12\) requires finding two numbers that multiply to \(-12\) and add up to \(-1\). These numbers are \(-4\) and \(+3\), which leads to the factorization \((x - 4)(x + 3)\).

By factoring these expressions, we are laying the groundwork for simplifying the problem, which helps in avoiding undefined behavior such as division by zero.

Simplifying Expressions

After factoring, the next step is simplifying the expression. Simplifying involves reducing the expression to its simplest form, which often means canceling out terms that are identical in both numerator and denominator. Simplification is crucial as it often prepares the expression for direct substitution.Once the polynomial is factored:

• The numerator \(x(x + 3)\) and the denominator \((x - 4)(x + 3)\) both include the common factor \((x + 3)\).
• This common factor can be canceled, simplifying the expression to \(\frac{x}{x - 4}\).

Simplification often allows for the evaluation of limits for values that previously might have led to an indeterminate form or division by zero.

Canceling Common Factors

Canceling common factors is a key step in simplifying expressions, especially in rational functions. By removing the common factors, you segregate the terms directly affecting the limit's behavior from those that do not. This step is vital because it removes potential undefined points that arise when the approach to the limit involves division by zero.In our example:

• The numerator \(x(x + 3)\) and the denominator \((x - 4)(x + 3)\) contain the common factor \((x + 3)\).
• By canceling \((x + 3)\), we avoid division by zero and any associated indeterminate forms.

Always be cautious about canceling. Ensure that the term you are canceling is indeed common and doesn't affect the point at which you're evaluating the limit.

Substitution in Limits

Substitution is typically one of the last steps in evaluating a limit, especially after an expression is simplified. When substituting in limits, you directly substitute the approaching value into the simplified expression. This allows for clear and easy calculations.In the problem at hand:

• After simplification to \(\frac{x}{x - 4}\), the next step is to evaluate \(\lim_{x \to -3}\).
• By substituting \(-3\) into the expression, we get \(\frac{-3}{-3 - 4} = \frac{-3}{-7} = \frac{3}{7}\).

The substitution reveals that the limit exists and is equal to \(\frac{3}{7}\). This underscores the effectiveness of proper simplification through accurate substitution, ensuring no indeterminate forms remain.