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Question
Select the correct answer from the given alternatives:
If f(x) = `((sin2x)tan5x)/("e"^(2x) - 1)^2`, for x ≠ 0 is continuous at x = 0, then f(0) is
Options
`10/"e"^2`
`10/"e"^4`
`5/4`
`5/2`
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Solution
`5/2`
Explanation;
f(x) is continuous at x = 0
f(0) = `lim_(x -> 0) "f"(x)`
= `lim_(x -> 0) ((sin2x)(tan5x))/("e"^(2x) - 1)^2`
= `(lim_(x -> 0)(sin2x)/(2x) xx lim_(x -> 0) (tan5x)/(5x) xx 2 xx 5)/((lim_(x -> 0) ("e"^(2x) - 1)/(2x)) xx (2)^2`
= `(1 xx 1 xx 2 xx 5)/((1)^2 xx 4) ...[(because x -> 0"," 2x -> 0"," 5x -> 0),("and" lim_(theta -> 0) sintheta/theta = 1"," lim_(theta -> 0) tantheta/theta = 1)]`
= `5/2`
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