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Question
Select the correct answer from the given alternatives.
`lim_(x -> 3) [(5^(x - 3) - 4^(x - 3))/(sin(x - 3))]` =
Options
log 5 – 4
`log 5/4`
`log5/log4`
`log5/4`
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Solution
`log 5/4`
Explanation;
`lim_(x -> 3) (5^(x - 3) - 4^(x - 3))/(sin(x - 3))`
Put x – 3 = h
∴ x = 3 + h
As → 3, h → 0
∴ Required limit
= `lim_("h" -> 0) (5^"h" - 4^"h")/(sin "h")`
= `lim_("h" -> 0) ((5^"h" - 1) - (4^"h" - 1))/sin"h"`
= `lim_("h" -> 0) (((5^"h" - 1))/"h" - ((4^"h" - 1))/"h")/(lim_("h" -> 0) sin"h"/"h"`
= `(log5 - log4)/1 ...[lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
= `log(5/4)`
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