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Question
Find k if the following function is continuous at the points indicated against them:
`f(x) = (1 + kx)^(1/x)` , for x ≠ 0
= `e^(3/2)` , for x = 0, at x = 0
Solution
f is continuous at x = 0
∴ `lim_(x→0) "f"(x)` = f(0)
∴ `lim_(x→0) (1 + "k"x)^(1/x) = "e"^(3/2)`
∴ `lim_(x→0) [(1 + "k"x)^(1/("k'"x)]]^"k" = "e"^(3/2)`
∴ `"e"^"k" = "e"^(3/2) ....[lim_(x→0) (1 + x)^(1/x) = "e"]`
∴ k = `3/2`
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