The given expression is:

$\frac{-\sin(x)}{x}$

This expression involves both trigonometric and rational functions. Its properties and behavior can be analyzed depending on the context, such as limits, derivatives, or integral evaluation. Here are a few important insights:

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1. As $x \to 0$:

Direct substitution into the function is undefined because $x$ is in the denominator. However, the limit can be analyzed using L'Hôpital's Rule:

$\lim_{x \to 0} \frac{-\sin(x)}{x}$

Differentiating the numerator and denominator:

[ \text{Numerator: } \frac{d}{dx}[-\sin(x)] = -\cos(x), \quad \text{Denominator: } \frac{d}{dx}[x] = 1 ]

So, the limit becomes:

$\lim_{x \to 0} \frac{-\sin(x)}{x} = \lim_{x \to 0} -\cos(x) = -1$

Thus:

$\lim_{x \to 0} \frac{-\sin(x)}{x} = -1$

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2. For Large $x$ ($x \to \infty$ or $x \to -\infty$):

As $x \to \infty$ or $x \to -\infty$, the numerator $-\sin(x)$ oscillates between -1 and 1, while $x$ grows without bound. This causes the fraction to approach 0:

$\lim_{x \to \pm\infty} \frac{-\sin(x)}{x} = 0$

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3. Derivative:

To compute the derivative of $f(x) = \frac{-\sin(x)}{x}$, we use the quotient rule:

[ f'(x) = \frac{\big(x \cdot \frac{d}{dx}[-\sin(x)] - [-\sin(x)] \cdot \frac{d}{dx}[x]\big)}{x^2} ]

$f'(x) = \frac{-x\cos(x) + \sin(x)}{x^2}$

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4. Integral:

The integral of $\frac{-\sin(x)}{x}$ does not have a simple closed-form solution. It is expressed using the special function called the Sine Integral ($\operatorname{Si}(x)$):

$\int \frac{-\sin(x)}{x} \, dx = -\operatorname{Si}(x) + C$

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5. Key Behavior Summary:

• The function is undefined at $x = 0$, but the limit exists ($-1$).
• It oscillates and decays towards zero as $|x| \to \infty$.
• It involves advanced mathematical tools for integral analysis.

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Would you like to explore:

1. A step-by-step application of L'Hôpital's Rule?
2. Plotting the function's graph for better visualization?
3. A deeper explanation of its integral?
4. Numerical approximations for specific $x$-values?
5. The oscillatory behavior of the numerator $-\sin(x)$?

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Tip: When analyzing functions involving limits, always test for cases where the denominator approaches zero or infinity to identify potential singularities or asymptotic behavior.