Question

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

Answer 1

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