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What is the value of limh0ln(1+h)h (with answer)

limh0ln(1+h)h=1 since:

(1) ln(a)ln(b)=ln(ab) since:

  • Let y=ln(a)ln(b)
  • ey=eln(a)ln(b)=eln(a)eln(b)=ab
  • y=ln(ab)

(2) limh0ln(x+h)ln(x)h=limh0ln(1+hx)h

(3)  Let t=hx so that h=xt and:

limh0ln(1+hx)h=limt0ln(1+t)xt=limt0(1xt)ln(1+t)

(4)  nln(a)=ln(an) since:

  • Let y=nln(a)
  • ey=enln(a)=(eln(a))n=an
  • y=ln(an)
(5)  limt0(1xt)ln(1+t)=limt0ln([1+t]1xt)=limt0ln([(1+t)1t]1x)=(1x)limt0ln([1+t]1t)

(6)  By definition, e=limn(1+1n)n so that if u=1t, then:

(1x)limt0ln([1+t]1t)=(1x)limuln([1+1u]u)=(1x)ln(e)=1x

(7)  Since limh0ln(x+h)ln(x)h=1x, it follows that:
limh0ln(1+h)h=limh0ln(1+h)ln(1)h=11=1

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