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Question
Evaluate the following limits:
`lim_(x -> oo)(1 + "k"/x)^("m"/x)`
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Solution
Let A = `lim_(x -> oo)(1 + "k"/x)^("m"/x)`
Put `"k"/x` = y
`"m"/x = "m"/"k" * y`
When x → ∞
We have y → 0
∴ A = `lim_(y -> 0) (1 + y)^("m"/"k" * y)`
log A = `log[lim_(y -> 0) (1 + y)^("m"/"k" * y)]`
= `lim_(y -> 0)[log(1 +y)^("m"/"k" * y)]`
= `lim_(y -> 0) ["m"/"k" * y log(1 + y)]`
= `"m"/"k" xx 0 xx log(1 + 0)`
log A = `"m/"k" xx 0 xx 0` = 0
A = e0
`lim_(x -> oo)(1 + "k"/x)^("m"/x)` = 1
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