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Question
Evaluate the following limit :
`lim_(x -> 0) [((25)^x - 2(5)^x + 1)/(x*sinx)]`
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Solution
`lim_(x -> 0) ((25)^x - 2(5)^x + 1)/(x*sinx)`
= `lim_(x -> 0) ((5^x)^2 - 2(5^x) + 1)/(x*sinx) ...[(25)^x = (5^2)^x = (5^x)^2]`
= `lim_(x -> 0) ((5^x - 1)^2)/(x*sinx)`
= `lim_(x -> 0) ((5^x - 1)^2/x^2)/((x*sinx)/x^2) ...[("Divide numerator and"),("denominator by" x^2.),(because x -> 0"," x ≠ 0),(therefore x^2 ≠ 0)]`
= `(lim_(x -> 0)((5^x - 1)/x)^2)/(lim_(x -> 0) sinx/x)`
= `(log5)^2/1 ...[∵ lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
= (log 5)2
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