Problem 383 - Calculus Volume 1

Question

For the following exercises, evaluate the limits with either L'Hôpital's rule or previously learned methods. $$ \lim _{x \rightarrow \infty} x \sin \left(\frac{1}{x}\right) $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Consider Simplifying the Expression

Begin by observing the expression \( x \sin\left( \frac{1}{x} \right) \). Notice that as \( x \to \infty \), \( \frac{1}{x} \to 0 \). This means we can look at \( \sin\left( \frac{1}{x} \right) \approx \frac{1}{x} \), since \( \sin(y) \approx y \) when \( y \) is close to 0.

Step 2: Substitute the Approximation

Replace \( \sin\left( \frac{1}{x} \right) \) with its approximation \( \frac{1}{x} \) to simplify the expression to \( x \cdot \frac{1}{x} = 1 \). Then the limit can be rewritten as: \[\lim _{x \rightarrow \infty} 1\].

Step 3: Evaluate the Simplified Limit

Since this simplification results in a constant function, the limit as \( x \to \infty \) is simply the constant value. The constant is \( 1 \), thus: \[\lim _{x \rightarrow \infty} x \sin \left(\frac{1}{x}\right) = 1\].

Key Concepts

Limits at Infinity

When you encounter limits at infinity in calculus, you are essentially looking at what happens to a function's value as the independent variable grows larger and larger. This is a fundamental idea in calculus because it helps us understand the behavior of functions under extreme conditions. In the original exercise, we have the limit \[\lim_{x \rightarrow \infty} x \sin \left(\frac{1}{x}\right)\]Here, as \(x\) approaches infinity, \(\frac{1}{x}\) approaches zero. This forms the basis for finding the limit. For many functions, examining their behavior as \(x\) approaches infinity can help predict long-term trends. This is critical for fields such as physics or economics, where predictions over large scales are necessary. Using approximations, as shown in the problem, where \(\sin(\frac{1}{x})\approx \frac{1}{x}\), allows us to simplify complex problems and achieve results effectively. Ultimately, limits at infinity help provide insight into the "end behavior" of functions.

Trigonometric Limits

Trigonometric limits involve limits that apply trigonometric functions like sine, cosine, and tangent. They often require special techniques in calculus because the values of trigonometric functions cycle and repeat. In the exercise, we deal with the limit of \(x \sin \left(\frac{1}{x}\right)\). As \(x\) becomes very large, \(\frac{1}{x}\) becomes very small, approaching zero. With small values around zero, sin functions can be approximated by their arguments or through linear approximations like \(\sin(y) \approx y\). This approximation helps transform the sine expression into a much simpler algebraic expression. By doing this substitution, we turn challenging trigonometric limits into more manageable problems. Understanding trigonometric limits is useful for more complex calculus problems, especially when dealing with wave-like behaviors or oscillations, which are common in real-world applications.

Calculus Problem Solving

Mastering calculus problem solving requires understanding different techniques and applying them to various problems effectively. One such technique is L'Hôpital's Rule, which is particularly useful when you identify indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).Though not directly used in this exercise, recognizing when to apply rules like L'Hôpital's can simplify solving limits significantly. For the specific problem \(\lim_{x \rightarrow \infty} x \sin \left( \frac{1}{x} \right)\), simplifying the expression with approximations, instead of L'Hôpital's Rule, was more efficient.When solving calculus problems:

• Look for patterns or approximations that might simplify the function.
• Apply the right rules or theorems for the situation at hand.
• Break down the problems into smaller steps, as this makes finding solutions easier.

Effective calculus problem solving relies on practice and familiarity with a broad range of techniques, making each new problem an opportunity to enhance your analytical skills.