Problem 26 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(21-26,\) find \(\lim _{x \rightarrow \infty} y\) and \(\lim _{x \rightarrow-\infty} y\). $$y=\frac{x \sin x+2 \sin x}{2 x^{2}}$$

Step-by-Step Solution

Answer

The limit as \(x\) approaches \(\infty\) or \(-\infty\) is undefined.

Step 1: Identify Limit Forms

First identify if the limit form is an indeterminate form of type \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). Substituting \(x=\infty\) and \(x=-\infty\) into the given function \(y=\frac{x \sin x+2 \sin x}{2 x^{2}}\) provides these forms.

Step 2: Apply L'Hospital's Rule once

Apply L'Hospital's rule by taking the derivative of the numerator and denominator. The derivative of the numerator \(xsinx + 2 sinx\) becomes, using product rule and constant rule, \( xsin(x) \to cosx - xsinx\) and \(2sinx \to 2cosx\). Therefore, the derivative of the complete numerator is \( cosx - xsinx + 2cosx\). The derivative of the denominator \(2x^2\) is \(4x\). The function becomes \(\frac{cosx - xsinx + 2cosx}{4x}\).

Step 3: Analyze new function

Check if the new function still has the limit form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). If it does, then apply L'Hospital's rule again. If not, then you may calculate the limit normally.

Step 4: Apply L'Hospital's Rule again

The application of L'Hospital's rule again gives us \(\frac{-sinx - sinx - xcosx + 2(-sinx)}{4}\). This given function simplifies to \(\frac{-3sinx - xcosx}{4}\). Note that as \(x -> \infty\) or \(x -> -\infty\), the sinusoidal terms will oscillate between -1 and 1.

Step 5: Compute the Limit

Now, compute the limit as \(x \rightarrow \infty\) and \(x\rightarrow-\infty\). As \(x\) approaches either \(\infty\) or \(-\infty\), \(-3sinx - xcosx\) will also oscillate between positive and negative values. The limit is therefore undefined as \(x\) tends to \(\pm\infty\).

Key Concepts

L'Hospital's Rule

L'Hospital's Rule is a handy tool in calculus for solving limits that initially present as indeterminate forms. It primarily helps when you encounter forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). To use it effectively:

• Differentiation is key. Differentiate the numerator and the denominator separately.
• Re-evaluate the limit with these new functions.
• If the new limit is still indeterminate, apply the rule again until a determinate form is reached.

This method relies heavily on derivatives, so a firm understanding of differentiation rules is essential. In the exercise, applying L'Hospital's Rule helped simplify the function, making it easier to analyze whether the limit approached a finite value or remained indeterminate.

Infinity Limits

When evaluating limits as \(x\) approaches infinity, it’s crucial to understand the behavior of the terms involved. An infinity limit explores how a function behaves as the input grows larger and larger:

• If a function continuously increases or decreases without bound, the limit might be infinite.
• For rational functions, compare the degrees of the numerator and the denominator to deduce the limit at infinity.

In the given exercise, the function's limit was investigated as \(x\) approached \(\infty\) and \(-\infty\). The trigonometric terms involved resulted in oscillation, ultimately leading to an undefined limit.

Trigonometric Functions

Trigonometric functions like \(\sin x\) and \(\cos x\) are periodic and oscillate between \(-1\) and \(1\). This means that without context, their behavior can be unpredictable over large ranges.

• Important properties: \(\sin x\) and \(\cos x\) oscillate; they don’t increase or decrease boundlessly.
• When combined with other functions, they can complicate limit analysis, often requiring tools like L'Hospital's Rule.

In the problem, both \(x \sin x\) and \(\cos x\) contribute to the oscillating behavior which complicates finding a finite limit as \(x\) approaches infinity. Understanding these functions' behaviors is key for solving related calculus problems.

Indeterminate Forms

Indeterminate forms occur when a direct substitution in a limit does not yield a specific value. Examples include \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), and others like \(0 \times \infty\) or \(1^\infty\). Addressing these forms typically requires additional techniques:

• Using algebraic manipulation to simplify the function.
• Applying L'Hospital's Rule to differentiate and simplify.
• Considering different mathematical theorems or transformations.

In the exercise, detecting indeterminate forms helped in identifying when and where to apply L'Hospital's Rule. Recognizing and transforming indeterminate forms is a fundamental skill in calculus, enabling students to accurately compute otherwise perplexing limits.