29. Let a1, a2, . . ., an be fixed real numbers and define a function f (x) = (x - a_1 ) (x - a_2 )...(x - a_n ) . What is  \lim_{x \rightarrow a _ 1 }  f (x) ? For some a \neq a _ 1 , a _ 2 .... a _n , compute l\lim_{ x \rightarrow a } f (x)

Answers (1)

Given,

f (x) = (x - a_1 ) (x - a_2 )...(x - a_n ) .

Now,

\\\lim_{x \rightarrow a _ 1 }f(x)=\lim_{x \rightarrow a _ 1 }[(x - a_1 ) (x - a_2 )...(x - a_n ) ]\\.=[\lim_{x \rightarrow a _ 1 }(x - a_1 )][\lim_{x \rightarrow a _ 1 }(x - a_2 )][\lim_{x \rightarrow a _ 1 }(x - a_n )] \\=0

Hence

\lim_{x \rightarrow a _ 1 }f(x)=0

Now,

\lim_{ x \rightarrow a } f (x)=\lim_{ x \rightarrow a } (x-a_1)(x-a_2)...(x-a_n)

\lim_{ x \rightarrow a } f (x)=(a-a_1)(a-a_2)(a-a_3)

Hence 

\lim_{ x \rightarrow a } f (x)=(a-a_1)(a-a_2)(a-a_3).

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