Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

find the indicated limit or state that it does not exist. In many cases, you will want to do some algebra before trying to evaluate the limit. $$ \lim _{w \rightarrow-2} \frac{(w+2)\left(w^{2}-w-6\right)}{w^{2}+4 w+4} $$

Step-by-Step Solution

Answer

The limit is -5.

Step 1: Factor the Numerator

The numerator of the given expression is \((w+2)(w^2 - w - 6)\). First, we need to factor \(w^2 - w - 6\). This can be factored as \((w-3)(w+2)\). So the numerator becomes \((w+2)((w-3)(w+2))\) or \((w+2)^2(w-3)\).

Step 2: Factor the Denominator

The denominator is \(w^2 + 4w + 4\). We recognize this as a perfect square trinomial, which factors as \((w+2)^2\).

Step 3: Simplify the Expression

Now we have the expression \(\frac{(w+2)^2(w-3)}{(w+2)^2}\). We can cancel the \((w+2)^2\) terms from the numerator and the denominator, simplifying the expression to \(w-3\).

Step 4: Evaluate the Limit

Now, we substitute \(w = -2\) into the simplified expression \(w-3\). That gives \(-2 - 3 = -5\). Therefore, the limit is \(-5\).

Key Concepts

Factoring Algebraic Expressions

When tackling algebraic expressions, especially for calculus problems like evaluating limits, factoring is an essential skill. Factoring simplifies the problem and reveals cancellation opportunities which often help in resolving indeterminate forms.Factoring involves breaking down a complex expression into a product of simpler ones. Let’s consider the expression \( w^2 - w - 6 \). To factor it, look for two numbers whose product is -6 (the constant term) and sum is -1 (the coefficient of \( w \)).For this example, you can use -3 and 2 as these numbers. Hence, the expression can be rewritten and factored as \((w-3)(w+2)\). This method helps you to identify the components that might be canceled out later in the simplification process, leading to a more straightforward calculation of the limit. Knowing how to identify perfect squares such as \( w^2 + 4w + 4 \), which factors into \((w+2)^2\), also comes under factoring skills. Recognizing these patterns not only expedites the process but also minimizes errors.

Simplifying Rational Expressions

Once the algebraic expressions are factored, the next crucial step is simplifying the rational expressions. This involves reducing the expression by canceling out common factors from both the numerator and the denominator.In our example, after factoring both parts of the expression, we obtain:

• Numerator: \((w+2)^2(w-3)\)
• Denominator: \((w+2)^2\)

Upon factoring, a common term \((w+2)^2\) is present in both the numerator and the denominator. You can cancel this term to simplify the rational expression to \(w-3\). This elimination is crucial as it further simplifies the calculation, allowing you to directly substitute the variable's limit value without running into indeterminate forms.Simplifying expressions not only makes subsequent calculations more manageable but can also help identify when the limit doesn’t exist, making this step integral in evaluating limits.

Evaluating Limits

Evaluating limits is a fundamental concept in calculus, typically used to find the output of a function as the input approaches a certain point. In this exercise, after simplifying, we focus on evaluating the limit of \( \lim_{w \to -2} (w-3) \).Now that the expression has been simplified to \(w-3\), you simply substitute \(w = -2\) into this function. So, you calculate:\[ w - 3 = -2 - 3 = -5. \]Therefore, the limit as \(w\) approaches -2 is \(-5\). Sometimes, direct substitution works straightforwardly, especially after simplification, because any potential undefined situations or problematic terms have already been addressed.Understanding how to evaluate limits after simplifying expressions is essential for solving various calculus problems, as it provides insights into the behavior of functions as they approach certain values.