Mathematics

Problem

29. Let

a_{1}, a_{2}, \ldots, a_{n}

be fixed real numbers and define a function

f(x)=\left(x-a_{1}\right)\left(x-a_{2}\right) \ldots\left(x-a_{n}\right)

. What is

\lim _{x \rightarrow a_{1}} f(x)

? For some

a \neq a_{1}, a_{2}, \ldots, a_{n}

, compute

\lim _{x \rightarrow a} f(x)

. a.

Intuitive explanation

The key insight is that $f(x)$ is a product of linear factors, and limits respect multiplication—you can evaluate each factor separately as $x$ approaches your target value. When $x \to a_1$ , one factor becomes zero while all others approach nonzero values, giving you zero. When $x \to a$ for any $a$ different from all the roots, every single factor approaches a nonzero constant, so their product is just that constant product.

Intuitive explanation

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