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Proof these results in limits

1. $\lim_{x\rightarrow0}(1+x)^{1/x} = e$

2. $\lim_{x\rightarrow0}(1+nx)^{1/x} = e^n$

3. $\lim_{x\rightarrow\infty }(1+\frac{1}{x})^{x} = e$

4. $\lim_{x\rightarrow\infty }(1+\frac{a}{x})^{x} = e^a$

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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)  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

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