if lim x→a [f(x)] g(x)
lim x→a f(x) = 1
and lim x→a g(x) = ∞
ie. 1∞ type indeterminate form
then lim x→a [f(x)] g(x) = e lim x→a g(x) [f(x) - 1] (when 1∞ type indeterminate form)
1) lim x→0 [1 + x] 1/x
1∞ type indeterminate form
so lim x→0 [1 + x] 1/x = e lim x→0 (1+x-1)/x = e1
2) lim x→0 [1 + nx] 1/x
1∞ type indeterminate form
so lim x→0 [1 + nx] 1/x = e lim x→0 (1+nx-1)/x = en
3) lim x→∞ [1 + 1/x] x
1∞ type indeterminate form
so lim x→∞ [1 + 1/x] x = e lim x→∞ (1+1/x-1) * x = e1
4) lim x→∞ [1 + a/x] x
1∞ type indeterminate form
so lim x→∞ [1 + a/x] x = e lim x→∞ (1+a/x-1) * x = ea