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Question
Select the correct answer from the given alternatives:
f(x) `{:(= ((16^x - 1)(9^x - 1))/((27^x - 1)(32^x - 1))",", "for" x ≠ 0),(= "k"",", "for" x = 0):}` is continuous at x = 0, then ‘k’ =
Options
`8/3`
`8/15`
`-8/15`
`20/3`
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Solution
`8/15`
Explanation:
f(x) is continuous at x = 0
`therefore "f"(0) = lim_(x -> 0) (x)`
`therefore "k" = lim_(x -> 0)((16^x - 1)(9^x - 1))/((27^x - 1)(32^x - 1))`
`= (lim_(x -> 0) ((16^x - 1)/x) xx lim_(x -> 0) ((9^x - 1)/x))/(lim_(x -> 0) ((27^x - 1)/x) xx lim_(x -> 0) ((32^x - 1)/x))`
`= (log 16 xx log 9)/(log 27 xx log 32) ...[because lim_(x -> 0) ("a"^x - 1)/x = log "a"]`
`= (4 log 2 xx 2 log 3)/(3 log 3 xx 5 log 2)`
`= (4 cancel(log 2) xx 2 cancel(log 3))/(3 cancel(log 3) xx 5 cancel(log 2))`
`= (4 xx 2)/(3 xx 5)`
`= 8/15`
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