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Question
Evaluate the following limits, if necessary use l’Hôpital Rule:
`lim_(x -> oo) (1 + 1/x)^x`
Solution
`lim_(x -> oo) (1 + 1/x)^x` .......`[1^oo "Indeterminate form"]`
Let `"g"(x) = (1 + 1/x)^x`
Taking log on both sides
log g(x) = `x log(1 + 1/x)`
`lim_(x -> oo) log "g"(x) = lim_(x -> oo) (log(1 + 1/x))/(1/x)` .......`[0/0 "Indeterminate form"]`
Applying L’ Hôpital’s Rule
= `lim_(x -> oo) [(1/(1 + 1/x) (- 1/x^2))/(- 1/x^2)]`
= `lim_(x -> oo) 1/(1 + 1/x)`
= 1
Exponentiating we get,
`lim_(x -> oo) "g"(x)` = e1 = e
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