Problem 44 - Calculus: Early Transcendentals

Question

Find the limit, if it exists. If the limit does not exist, explain why. \( \displaystyle \lim_{x \to -2}\frac{2 - |x|}{2 + x} \)

Step-by-Step Solution

Answer

The limit exists and is 1.

Step 1: Understand Function Components

Analyze the function \( \frac{2 - |x|}{2 + x} \). The absolute value \(|x|\) impacts the function differently based on whether \(x\) is negative or positive. Near \(x = -2\), \(|x| = -x\), since \(-2\) is negative.

Step 2: Simplify the Expression Near \(x = -2\)

Substitute \(|x| = -x\) into the function for \(x\) near \(-2\). The result is \(f(x) = \frac{2 - (-x)}{2 + x} = \frac{2 + x}{2 + x}\).

Step 3: Evaluate Limit of Simplified Expression

Simplify \(f(x) = \frac{2 + x}{2 + x}\) to get \(1\) for all \(x eq -2\). As \(x\) approaches \(-2\), the expression is undefined at \(x = -2\), but the limit can be evaluated.

Step 4: Conclude with the Limit

Because \(f(x) = 1\) for all \(x eq -2\) in the neighborhood of \(x = -2\), \( \lim_{x \to -2} \frac{2 - |x|}{2 + x} = 1\). The limit exists since the undefined condition does not affect the neighborhood behavior.

Key Concepts

Understanding the Absolute Value Function

The absolute value function is crucial when dealing with limits, particularly in expressions involving absolute values. The absolute value of a number, symbolized as \(|x|\), refers to its non-negative magnitude. This function affects our operations because it changes based on whether the input number is positive or negative.
For example, \(|x| = x\) if \(x \gt 0\) and \(|x| = -x\) if \(x \lt 0\). This property moderates which expression to use when simplifying or evaluating limits.
In the exercise, as we approach \(x = -2\), \(|x|\) will behave as \(-x\) since \(-2\) is less than zero. Recognizing the behavior of absolute value functions based on the input's sign is important for correctly setting up limit problems.

Simplifying Expressions

Simplifying expressions is a fundamental step when working with calculus limits, especially with absolute values. Simplification involves rewriting expressions in simpler or more recognizable forms, often to eliminate complex parts or make calculations easier.
In our exercise, after determining how the absolute value behaves near \(-2\), the expression \(2 - |x|\) becomes \(2 + x\). This observation transforms the initial expression \(\frac{2 - |x|}{2 + x}\) into a much simpler version: \(\frac{2 + x}{2 + x}\).
This simplification is crucial because it transforms the problem into an easier format to understand and solve. The expression simplifies to \(1\) for all \(x eq -2\), showcasing how simplification can reveal the steady behavior of a function near specific points.

Evaluating Limits

Evaluating limits is about determining the value that a function approaches as the variable gets closer to a particular point. This concept is central to calculus and helps find function behavior near points where the function might not be well-defined.
In our exercise, we wanted to see how the function \(\frac{2 - |x|}{2 + x}\) behaves as \(x\) approaches \(-2\). Since simplification showed \(\frac{2 + x}{2 + x}\) equals \(1\) for \(x eq -2\), we understand that although the original expression is undefined at \(-2\), the behavior around this point is consistent and equals \(1\).
Thus, we can conclude that the limit exists, and therefore, \(\lim_{x \to -2} \frac{2 - |x|}{2 + x} = 1\). Knowing how to approach such evaluations allows breaking down more complex calculus problems with confidence.