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Question
Evaluate the following Limits: `lim_(x -> 0)[((5^x - 1)^2)/(x*log(1 + x))]`
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Solution
`lim_(x -> 0)((5^x - 1)^2)/(x*log(1 + x))`
= `lim_(x -> 0) ((5^x - 1)^2/x^2)/((x*log(1 + x))/x^2) ...[("As" x -> 0"," x ≠ 0 therefore x^2 ≠ 0),("Divide Numerator and"),("Denominator by " x^2)]`
= `(lim_(x -> 0)((5^x - 1)/x)^2)/(lim_(x -> 0)(log(1 + x))/x`
= `(log 5)^2/1 ...[(lim_(x -> 0) ("a"^x - 1)/x = log "a"","),(lim_(x -> 0) (log(1 + x))/x = 1)]`
= (log 5)2
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