Let xnβ=n1β,nβZ+
xnββ0+limβ(1+xnβ)xnβ1β=nββlimβ(1+n1β)n Use Newton's Binomial formula
(1+n1β)n=1+nβ
n1β+1β
2n(nβ1)ββ
n21β+
+1β
2β
3n(nβ1)(nβ2)ββ
n31β+...+1β
2β
...β
nn(nβ1)...(1)ββ
nn1β>1+1=2
Let f(x)=ln(x), then fβ²(x)=1/x. Thus fβ²(1)=1/1=1.
From the definition of a derivative as a limit, we have
fβ²(1)=hβ0limβhf(1+h)βf(1)β=
=xβ0limβxf(1+x)βf(1)β=xβ0limβxln(1+x)βln(1)β=
=xβ0limβxln(1+x)β=xβ0limβln(1+x)x1β Because fβ²(1)=1, we have
xβ0limβln(1+x)x1β=1Theorem
If f is continuous at b and xβalimβg(x)=b, then xβalimβf(g(x))=f(b)
xβalimβf(g(x))=f(xβalimβg(x)) Since the exponential function is continuous, we have
e=e1=exβ0limβln(1+x)x1β=xβ0limβeln(1+x)x1β=
=xβ0limβ(1+x)x1β Therefore
xβ0limβ(1+x)x1β=e
xβ0+limβ(1+x)x1β=e
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