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Question Number 99936 by student work last updated on 24/Jun/20

lim_(x→0) (((1+x)^k −1)/x)=? help me

$$\mathrm{li}\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{m}}\frac{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{k}} −\mathrm{1}}{\mathrm{x}}=? \\ $$$$\mathrm{help}\:\mathrm{me} \\ $$

Commented by bemath last updated on 24/Jun/20

for k > 0 ⇒ lim_(x→0) ((k(1+x)^(k−1) .1)/1) = k

$$\mathrm{for}\:\mathrm{k}\:>\:\mathrm{0}\:\Rightarrow\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{k}\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{k}−\mathrm{1}} .\mathrm{1}}{\mathrm{1}}\:=\:\mathrm{k} \\ $$

Commented by Dwaipayan Shikari last updated on 24/Jun/20

lim_(x→0) (((1+x)^k −1)/(1+x−1))=k.1=k

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left(\mathrm{1}+{x}\right)^{{k}} −\mathrm{1}}{\mathrm{1}+{x}−\mathrm{1}}={k}.\mathrm{1}={k} \\ $$

Commented by Dwaipayan Shikari last updated on 24/Jun/20

k

$${k} \\ $$

Answered by mathmax by abdo last updated on 24/Jun/20

take a try first and dont waste your time ...

$$\mathrm{take}\:\mathrm{a}\:\mathrm{try}\:\mathrm{first}\:\mathrm{and}\:\mathrm{dont}\:\mathrm{waste}\:\mathrm{your}\:\mathrm{time}\:... \\ $$

Answered by mathmax by abdo last updated on 24/Jun/20

f(x) =(((1+x)^n −1)/x) ⇒f(x) =((Σ_(k=0) ^n C_n ^k x^k −1)/x) =((Σ_(k=1) ^n C_n ^k x^k )/x) =Σ_(k=1) ^n C_n ^k x^(k−1) =Σ_(k=0) ^(n−1) C_n ^(k+1) x^k =C_n ^1 +C_n ^2 x +C_n ^3 x^2 +... +C_n ^n x^n ⇒lim_(x→0) f(x) =C_n ^1 =n

$$\mathrm{f}\left(\mathrm{x}\right)\:=\frac{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{n}} −\mathrm{1}}{\mathrm{x}}\:\Rightarrow\mathrm{f}\left(\mathrm{x}\right)\:=\frac{\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:\mathrm{x}^{\mathrm{k}} −\mathrm{1}}{\mathrm{x}}\:=\frac{\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:\mathrm{x}^{\mathrm{k}} }{\mathrm{x}}\:=\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:\mathrm{x}^{\mathrm{k}−\mathrm{1}} \\ $$$$=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}−\mathrm{1}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}+\mathrm{1}} \:\mathrm{x}^{\mathrm{k}} \:=\mathrm{C}_{\mathrm{n}} ^{\mathrm{1}} \:+\mathrm{C}_{\mathrm{n}} ^{\mathrm{2}} \:\mathrm{x}\:+\mathrm{C}_{\mathrm{n}} ^{\mathrm{3}} \:\mathrm{x}^{\mathrm{2}} \:+...\:+\mathrm{C}_{\mathrm{n}} ^{\mathrm{n}} \:\mathrm{x}^{\mathrm{n}} \:\Rightarrow\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \mathrm{f}\left(\mathrm{x}\right)\:=\mathrm{C}_{\mathrm{n}} ^{\mathrm{1}} \:=\mathrm{n} \\ $$

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