Question

Let a1, a2,..., an be fixed real numbers and define a function f(x)=(x-a1)(x-a2)...(x-an).
What is $\lim_{x\rightarrow a_1}$ f(x)?
For some a ≠ a1, a2 .....an, Compute $\lim_{x\rightarrow a}$ f(x).

Answer 1

The given function is f(x) = (x − a1) ( x − a2)... ( x − an)
$\lim_{x\rightarrow a_1}$ f(x)= $\lim_{x\rightarrow a_1}$ [(x-a1)(x − a2).....(x − an)]
=[$\lim_{x\rightarrow a_1}$ (x − a1) ][$\lim_{x\rightarrow a_1}$(x − a2)]..... [$\lim_{x\rightarrow a_1}$ (x-an)]
=(a1-a1)(a1-a2)....(a1 -an) = 0
∴$\lim_{x\rightarrow a_1}$ f(x)=0
Now, $\lim_{x\rightarrow a}$ f(x)= $\lim_{x\rightarrow a}$[(x − a1)(x-a2)...(x-an)]
= [$\lim_{x\rightarrow (x-a_1)}$][$\lim_{x\rightarrow a}$ (x-a2)]...$\lim_{x\rightarrow a}$(x-an)]
=(a-a1) (a-a2)....(a-an)
∴$\lim_{x\rightarrow a}$ f(x)=(a-a1)(a-a2)...(a-an)