Question - Solving Limit Using Sine Addition Formula
Solution:
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 $$ \cos(a) $$.
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