Solving Limit Using Sine Addition Formula
The limit provided in the image is:
\[ \lim_{x \to a} \frac{\sin x - \sin a}{x - a} \]
We can solve this limit by using the Sine Addition Formula in the form of the difference:
\[ \sin x - \sin a = 2 \cos\left(\frac{x + a}{2}\right) \sin\left(\frac{x - a}{2}\right) \]
Now let's apply this transformation and compute the limit:
\[ \lim_{x \to a} \frac{2 \cos\left(\frac{x + a}{2}\right) \sin\left(\frac{x - a}{2}\right)}{x - a} = \lim_{x \to a} \left(2 \cos\left(\frac{x + a}{2}\right)\right) \cdot \left(\frac{\sin\left(\frac{x - a}{2}\right)}{\frac{x - a}{2}}\right) \]
As \( x \) approaches \( a \), the term \( \frac{x - a}{2} \) approaches 0. Thus, the second factor:
\[ \lim_{x \to a} \frac{\sin\left(\frac{x - a}{2}\right)}{\frac{x - a}{2}} \]
approaches 1, because it's the definition of the derivative of sine at 0:
\[ \lim_{h \to 0} \frac{\sin(h)}{h} = 1 \]
Also, since \( \cos\left(\frac{x + a}{2}\right) \) is continuous, we can simply replace \( x \) with \( a \):
\[ \lim_{x \to a} \cos\left(\frac{x + a}{2}\right) = \cos\left(\frac{2a}{2}\right) = \cos(a) \]
Multiplying these two results yields:
\[ \cos(a) \cdot 1 = \cos(a) \]
Therefore, the value of the limit is \( \cos(a) \).
