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Question
Select the correct answer from the given alternatives:
If f(x) = `((4 + 5x)/(4 - 7x))^(4/x)`, for x ≠ 0 and f(0) = k, is continuous at x = 0, then k is
Options
e7
e3
e12
`"e"^(3/4)`
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Solution
e12
Explanation;
f(x) is continuous at x = 0
∴ f(0) = `lim_(x -> 0) "f"(x)`
= `lim_(x -> 0) ((4 + 5x)/(4 - 7x))^(4/x)`
= `lim_(x -> 0) [(4(1 + (5x)/4))/(4(1 - (7x)/4))]^(4/x)`
= `(lim_(x -> 0) [(1 + (5x)/4)^(4/(5x))]^5)/(lim_(x -> 0)[(1 - (7x)/4)^((-4)/(7x))]^(-7))`
= `"e"^5/"e"^(-7) ...[(because x -> 0"," (5x)/4 -> 0"," (-7x)/4 -> 0),("and" lim_(x -> 0) (1 + x)^(1/x) = "e")]`
= e12
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