If $$F(x) = \sqrt {9 - {x^2}} $$, then what is $$\mathop {\lim }\limits_{x \to 1} {{F(x) - F(1)} \over {x - 1}}$$ equal to ?

Let f(x + y) = f(x) f(y) for all x and y. Then, what is f'(5) equal to [where f' (x) is the derivative of f(x)]?

Let f(x) be defined as follows

$$f(x) = \left\{ {\matrix{ {2x + 1,} & { - 3 < x < - 2} \cr {x - 1,} & { - 2 \le x < 0} \cr {x + 2,} & {0 \le x < 1} \cr } } \right.$$

Which one of the following statements is correct in respect of the above function?

Consider the following statements

1. If $$\mathop {\lim }\limits_{x \to a} f(x)$$ and $$\mathop {\lim }\limits_{x \to a} g(x)$$ both exist, then $$\mathop {\lim }\limits_{x \to a} \{ f(x)g(x)\} $$ exists.

2. If $$\mathop {\lim }\limits_{x \to a} \{ f(x)g(x)\} $$ exists, then both $$\mathop {\lim }\limits_{x \to a} f(x)$$ and $$\mathop {\lim }\limits_{x \to a} g(x)$$ must exist.

Which of the above statement is/are correct?