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Question
Evaluate the following limit :
`lim_(x -> 0)[("a"^x + "b"^x + "c"^x - 3)/sinx]`
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Solution
`lim_(x -> 0)("a"^x + "b"^x + "c"^x - 3)/sinx`
= `lim_(x -> 0) (("a"^x - 1) + ("b"^x - 1) + ("c"^x - 1))/sinx`
= `lim_(x -> 0) (("a"^x - 1 + "b"^x - 1 + "c"^x - 1)/x)/(sinx/x) ...[("Divide numerator and"),("denominator by" x.),(because x -> 0 therefore x ≠ 0)]`
= `lim_(x -> 0) ((("a"^x - 1)/x) + (("b"^x - 1)/x) + (("c"^x - 1)/x))/(sinx/x)`
= `((lim_(x -> 0) ("a"^"x" - 1)/x) + (lim_(x -> 0) ("b"^x - 1)/x) + (lim_(x -> 0) ("c"^x - 1)/x))/((lim_(x -> 0) sinx/x))`
= `(log"a" + log"b" + log"c")/1 ...[because lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
= log (abc)
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