Problem 50 - Essential Calculus

Question

Use the series representation of the function \(f\) to find \(\lim _{x \rightarrow 0} f(x)\) (if it exists). $$ f(x)=\frac{\sin x}{x} $$

Step-by-Step Solution

Answer

The limit as \(x\) approaches 0 of \(f(x)=\frac{\sin x}{x}\) is 1.

Step 1: Substituting the Maclaurin series representation for sin(x)

Let's substitute the series representation of \(\sin x\) into the function. We get: \( f(x) = \frac{x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \cdots\)

Step 2: Find the limit

Now, let's take the limit as \(x\) approaches 0: \( \lim_{x \rightarrow 0} f(x) =\lim_{x \rightarrow 0} (1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \cdots)\)

Step 3: Substitute x=0 into the function

If we substitute \(x=0\) into the function we get: \( \lim_{x \rightarrow 0} f(x) = 1 - \frac{0}{3!} + \frac{0}{5!} - \frac{0}{7!} + \cdots = 1 \).

Key Concepts

Maclaurin series

The Maclaurin series is a special case of the Taylor series, which provides a way to represent functions as infinite sums of terms calculated from the function's derivatives at a single point. Specifically, the Maclaurin series uses the point \(x = 0\). The benefit of using such a series is that it simplifies complex trigonometric functions like sine and cosine, making them easier to work with in calculus.When we use the Maclaurin series for \(\sin x\), we get:\[\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\]This representation converges to \(\sin x\) for all values of \(x\). The series effectively breaks down what can be a difficult-to-calculate function into an infinite set of simpler polynomial terms.Using the Maclaurin series helps us analyze limits and other calculus-based operations by simplifying the function into a sum of known components.

• Simplifies the study of functions.
• Transforms functions into polynomial forms.
• Makes calculation of limits easier.

Series representation

Series representation is a powerful mathematical tool that expresses a function as an infinite sum of terms. The key advantage of this method is that complex functions can be broken down into a simple series of operations, often making them easier to handle and analyze.For example, in the original exercise, the function \(f(x) = \frac{\sin x}{x}\) is tricky in its original form. By using the series representation of \(\sin x\), the problem becomes more manageable. By substituting the series from the Maclaurin expansion, the function simplifies to:\[f(x) = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \cdots\]This simplification highlights the usefulness of series representation when finding limits or derivatives.Series representations are crucial for solving calculus problems, especially when direct computation is cumbersome. They provide a foundational technique in mathematical analysis.

• Transforms complex functions into simpler series.
• Allows for easier computation of derivatives and limits.
• Forms the basis of many numerical methods.

Sin function

The sine function, denoted as \(\sin x\), is one of the fundamental trigonometric functions. It is periodic, meaning it repeats values in regular intervals, and is defined in the context of right-angled triangles and unit circles. The function is often introduced in terms of its geometric interpretation, but in calculus, the focus shifts to its analytical expression.The sine function is uniquely defined by its infinite series expansion through the Maclaurin series:\[\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\]This expression allows it to be computed for both small and large values of \(x\). It is especially useful in finding limits as \(x\) approaches 0, as seen in the original exercise. By using the series, it becomes apparent that \(\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1\), directly from the pattern of the series.Understanding the sine function's series expansion is crucial for grasping more advanced math concepts. The trigonometric identity established helps in improving the calculation speed and providing accurate results.

• A key function in trigonometry and calculus.
• Defined by its periodic nature and series expansion.
• Vital for solving limits and other advanced mathematical problems.