Problem 31 - Calculus Single Variable

Question

Use an algebraic manipulation to put the limit in a form which can be treated using l'Hôpital's Rule; then evaluate the limit. \(\lim _{x \rightarrow-2}(x+2) \tan (\pi x / 4)\)

Step-by-Step Solution

Answer

The limit does not converge to a finite value; it diverges.

Step 1: Identify Indeterminate Form

First, we substitute \(x = -2\) into the function \((x+2) \tan(\pi x / 4)\) to determine if the limit results in an indeterminate form. Substituting, we get:\((x+2) = (-2+2) = 0\)\(\tan(\pi x / 4) = \tan(\pi(-2) / 4) = \tan(-\pi/2)\), which is undefined.Thus, the expression involves a \(0 \times \text{undefined}\) form. We can express the tangent function near these values to identify the \(0/0\) form needed to apply L'Hôpital's Rule.

Step 2: Rewrite Limit for L'Hôpital's Rule

To bring this limit to \(0/0\) form, rewrite the function:\[ \frac{\tan(\pi x / 4)}{1/(x+2)} \]This new expression is of the form \( \frac{0}{0} \) as \(x \to -2\), allowing us to apply L'Hôpital's Rule.

Step 3: Differentiate Numerator and Denominator

Apply L'Hôpital's Rule by differentiating the numerator and the denominator. The derivatives are:- Numerator: Derivative of \(\tan(\pi x / 4)\) with respect to \(x\) is: \[ \frac{d}{dx} \tan(\pi x / 4) = \sec^2(\pi x / 4) \cdot \frac{\pi}{4} \]- Denominator: Derivative of \(1/(x+2)\) with respect to \(x\) is: \[ \frac{d}{dx} (1/(x+2)) = -\frac{1}{(x+2)^2} \]

Step 4: Apply L'Hôpital's Rule

Applying the rule, the limit becomes:\[ \lim_{x \to -2} \frac{\sec^2(\pi x / 4) \cdot \frac{\pi}{4}}{-1/(x+2)^2} \]This simplifies to:\[ \lim_{x \to -2} \frac{-\pi}{4} \sec^2(\pi x / 4) \cdot (x+2)^2 \]

Step 5: Evaluate Limit

Substitute \(x = -2\) again into the simplified expression:- \(\sec^2(\pi x/4) = \sec^2(-\pi/2)\), which behaves as \(1/\cos^2(-\pi/2)\) and approaches infinity.Thus, the expression appears inconsistent or diverging to infinity rather than providing a finite limit.

Key Concepts

Limit evaluation

Limit evaluation is a fundamental technique in calculus, particularly useful when dealing with functions that become undefined or produce indeterminate forms when directly evaluated at a certain point.
In this exercise, our task was to evaluate the limit \( \lim _{x \rightarrow-2}(x+2) \tan (\pi x / 4) \). To tackle this, we must consider the behavior of the function as \(x\) approaches -2. Simply substituting \(x = -2\) resulted in a product \(0 \times \text{undefined}\), indicating we cannot evaluate the limit directly.
In such cases, mathematical tools and techniques such as algebraic manipulation and calculus rules like L'Hôpital’s Rule come in handy. We can manipulate the equation to transform it into a form suitable for further analysis by applying these rules.

• Recognizing indeterminate forms (like \(0/0\)) allows us to recast the function into a standard limit problem.
• Using calculus rules helps further to differentiate and simplify into manageable calculations.

This systemic approach turns complex appearences into a solvable sequence of calculations, as seen by applying L'Hôpital's Rule in later steps.

Indeterminate forms

Indeterminate forms are a key concept when it comes to evaluating limits, especially in calculus. They occur when substituting values directly into a function results in forms like \(0/0\), \(\infty/\infty\), \(0\times\infty\), and others that need resolution before the limit can be evaluated. In our example, substituting \(x = -2\) into the function \((x+2) \tan(\pi x / 4)\) at the beginning leads us to the unwanted form of \(0 \times \text{undefined}\), which is not directly solvable.

To handle such scenarios:

• Look for a way to re-express the function into a form where these difficulties disappear, often by involving algebraic simplification or rearrangement.
• Once rewired into a manageable indeterminate form such as \(0/0\), calculus tools like L'Hôpital's Rule can take over.

Successfully identifying and manipulating indeterminate forms is crucial in calculus, allowing us to reveal the actual behavior or trend of a function as it approaches a particular input.

Differentiation concepts

Differentiation is a cornerstone in calculus, crucial for transforming complex limit problems into simpler forms. In this exercise, after recognizing the indeterminate form \(0/0\), we use L'Hôpital’s Rule, which employs differentiation to solve such limits. This involves differentiating the numerator and the denominator of our expression separately.

In applying L'Hôpital’s Rule here, we:

• First differentiate the numerator \(\tan(\pi x / 4)\) to get \(\sec^2(\pi x / 4) \cdot \frac{\pi}{4}\).
• Then differentiate the denominator \(1/(x+2)\) to get \(-1/(x+2)^2\).

Through this process of differentiation, the original limit is transformed from a problematic indeterminate form into an expression that approaches a recognizable mathematical behavior. Differentiation in this context becomes a powerful tool not only to simplify algebraic expressions but also to understand a function's behavior and its eventual trend. It confirms how calculus can unravel sophisticated mathematical problems, enhancing overall comprehension and applicability of mathematics.