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Question

Let a1,a2,,an be fixed real numbers and define a function f(x)=(xa1)(xa2)(xan)
What is limxa1f(x)? For some aa1,a2.an compute limxaf(x).


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Solution

Given : f(x)=(xa1)(xa2)(xan)
limxa1f(x)=limxa1(xa1)(xa2)(xan)
=(a1a1)(a1a2)(a1an)
=0×(a1a2)(a1an)
=0
Hence, limxa1f(x)=0

Now,
limxaf(x)=limxa(xa1)(xa2)(xan)
Substituting x=a
=(aa1)(aa2)(aan)
Hence, limxaf(x)=(aa1)(aa2)(aan)

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