lim x -> 0 (sin(1/2 * pi + x) - sin(x))/x

To solve the limit

\[
\lim_{x \to 0} \frac{\sin\left(\frac{\pi}{2} + x\right) - \sin(x)}{x}
\]

we'll start by using the trigonometric identity for the sine of a sum:

\[
\sin\left(\frac{\pi}{2} + x\right) = \sin\left(\frac{\pi}{2}\right)\cos(x) + \cos\left(\frac{\pi}{2}\right)\sin(x)
\]

Since \(\sin\left(\frac{\pi}{2}\right) = 1\) and \(\cos\left(\frac{\pi}{2}\right) = 0\), the expression simplifies to:

\[
\sin\left(\frac{\pi}{2} + x\right) = \cos(x)
\]

Thus, the original limit becomes:

\[
\lim_{x \to 0} \frac{\cos(x) - \sin(x)}{x}
\]

To resolve the indeterminate form \(\frac{0}{0}\) as \(x \to 0\), we use L'Hôpital's Rule. According to the rule:

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

Evaluating at \(x = 0\):

\[
-\sin(0) - \cos(0) = -0 - 1 = -1
\]

Thus, the limit is:

\[
\boxed{-1}
\]

Would you like to explore more about this problem or any related topics?

Here are some related questions to think about:
1. What are other applications of L'Hôpital's Rule?
2. How does the Taylor series approximation help in evaluating limits?
3. What is the geometric interpretation of

Learn how to evaluate the limit lim x -> 0 (sin(pi/2 + x) - sin(x))/x with a step-by-step solution. This problem involves using trigonometric identities and applying L'Hôpital's Rule to resolve the indeterminate form. Suitable for advanced high school students and anyone interested in calculus and trigonometry.

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