Questions: lim as h approaches 0 of (sin((7π/6)+h)+1/2)/h

Solution Steps

To solve this limit problem, we can use the derivative definition. The expression resembles the derivative of a function at a point. Specifically, it looks like the derivative of \( f(x) = \sin(x) \) at \( x = \frac{7\pi}{6} \). We can use the derivative of the sine function, which is \( \cos(x) \), to find the limit.

We need to evaluate the limit

\[ \lim_{h \rightarrow 0} \frac{\sin\left(\frac{7\pi}{6} + h\right) + \frac{1}{2}}{h}. \]

This expression resembles the definition of the derivative of a function at a point.

The limit can be interpreted as the derivative of the function

\[ f(x) = \sin(x) + \frac{1}{2} \]

at the point \( x = \frac{7\pi}{6} \). The derivative of \( f(x) \) is given by

\[ f'(x) = \cos(x). \]

Now, we calculate the derivative at the specific point:

\[ f'\left(\frac{7\pi}{6}\right) = \cos\left(\frac{7\pi}{6}\right). \]

Using the known value of \( \cos\left(\frac{7\pi}{6}\right) \), we find:

\[ \cos\left(\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2} \approx -0.8660. \]

Final Answer