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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 -> a_1) f(x)` ? For some a ≠ a1, a2, ..., an, compute `lim_(x -> a) f(x)`
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Solution
To factor (x − a1),
If x → a1, x − a1 → 0
∴ `lim_(x - a_1) (x - a_2) = (a_1 - a_2)`
f(x) = (x – a1) (x – a2) ….. (x – an)
`lim_(x - a_1) f(x) = lim_(x - a_1)` (x – a1) (x – a2) ….. (x – an)
= `lim_(x - a_1) (x - a_1) lim_(x - a_1) (x- a_2) (x - a_3) ...... (x - a_n)`
= 0 × (a1 − a2) (a1 − a3) .......(a − an) 0
When, a ≠ a1, a2 ........, an
As soon as x → a, x → a1 → a − a1
a − a1 is neither zero nor undefined.
Thus, the values of the second factor will be a − a2, a − a3 ....., a - an.
Hence, `lim_(x - a) f(x) = lim_(x - a)` (x – a1) (x – a2) ….. (x – an)
= (a – a1) (a – a2) ….. (a – an)
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