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When you encounter a function that appears to have a division by zero—like when finding the limit of \( \frac{\sin x}{x} \) as \( x \rightarrow 0 \)—L'Hopital's Rule is a powerful technique. This rule is applicable when both the numerator and the denominator approach 0 or ±∞ as \( x \) approaches a certain value. L'Hopital's Rule states that the limit of the function can be evaluated by taking the derivatives of the numerator and denominator separately. For our specific function:

• The numerator is \( \sin x \) and its derivative is \( \cos x \).
• The denominator is \( x \) with a derivative of 1.

So, the limit becomes \( \frac{\cos x}{1} \), and evaluating this at \( x = 0 \) gives us \( \cos 0 = 1 \). Thus, L'Hopital's Rule confirms that the function \( \frac{\sin x}{x} \) approaches 1 as \( x \rightarrow 0 \). This makes the function well-behaved everywhere, even at \( x = 0 \), eliminating any concern of an actual singularity.

The sine function is one of the fundamental trigonometric functions, and it's extensively used in mathematics. Its graph represents the classic wave-like pattern that oscillates endlessly through the x-axis.
The Taylor Series expansion is a method of expressing functions as an infinite sum of terms calculated from the values of its derivatives at a single point.
For the sine function,

• The Taylor Series expansion at \( x = 0 \) is given by: \( \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \).
• More generally, this representation can be written as \( \sin x = \sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k+1)!} x^{2k+1} \).

Each term in this series alternates between positive and negative, converging quickly to the actual sine function as more terms are added.

Series representation is a way to express functions like \( \sin x \) as an infinite sum of easier-to-evaluate polynomial terms. This is especially useful when dealing with complex functions, as series can simplify calculations by approximating the function.
In our exercise, the goal was to find the series representation of \( \frac{\sin x}{x} \). By using the known series representation of \( \sin x \),

• We divide each term of the sine series by \( x \), resulting in: \( \sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k+1)!} x^{2k-1} \).
• This means, for \( k = 0 \), the term simplifies to 1, confirming it yields the expected value of the function at \( x = 0 \).

This series representation becomes extremely effective for evaluating the function near \( x = 0 \) and demonstrates how each term is crafted by dividing the sine series by \( x \), enhancing the function's applicability in various mathematical contexts.