Let $a_{1}, a_{2}, \dots, a_{n}$ be fixed real numbers and define a function $f(x) = (x - a_{1})(x - a_{2}) \dots (x - a_{n})$. What is $\lim_{x \to a_{1}} f(x)$? For some $a \neq a_{1}, a_{2}, \dots, a_{n}$,compute $\lim_{x \to a} f(x)$.

The given function is $f(x) = (x - a_{1})(x - a_{2}) \dots (x - a_{n})$.
$\lim_{x \to a_{1}} f(x) = \lim_{x \to a_{1}} [(x - a_{1})(x - a_{2}) \dots (x - a_{n})]$
$= (a_{1} - a_{1})(a_{1} - a_{2}) \dots (a_{1} - a_{n}) = 0 \times (a_{1} - a_{2}) \dots (a_{1} - a_{n}) = 0$.
Therefore,$\lim_{x \to a_{1}} f(x) = 0$.
Now,for $a \neq a_{1}, a_{2}, \dots, a_{n}$,
$\lim_{x \to a} f(x) = \lim_{x \to a} [(x - a_{1})(x - a_{2}) \dots (x - a_{n})]$
$= (a - a_{1})(a - a_{2}) \dots (a - a_{n})$.
Therefore,$\lim_{x \to a} f(x) = (a - a_{1})(a - a_{2}) \dots (a - a_{n})$.