Question-165698

Question Number 165698 by azizmathhacker last updated on 06/Feb/22

Answered by alephzero last updated on 06/Feb/22

lim_(x→∞) e^(ln ((x+1)/x)) = e^(ln 1) = 1

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}{e}^{\mathrm{ln}\:\frac{{x}+\mathrm{1}}{{x}}} \:=\:{e}^{\mathrm{ln}\:\mathrm{1}} \:=\:\mathrm{1} \\ $$

Commented by chrisbridge last updated on 06/Feb/22

lim e^(ln ((x+1)/x) = lim ((x+1)/x) = (x/x) + (1/x)) x →∞ = 1 + (1/x) = 1 +(1/∞) = 1 + 0 = 1

$${lim}\:{e}^{\mathrm{ln}\:\frac{{x}+\mathrm{1}}{{x}}\:=\:\:{lim}\:\frac{{x}+\mathrm{1}}{{x}}\:=\:\frac{{x}}{{x}}\:+\:\frac{\mathrm{1}}{{x}}} \\ $$$${x}\:\rightarrow\infty \\ $$$$\:\:=\:\mathrm{1}\:+\:\frac{\mathrm{1}}{{x}}\:=\:\mathrm{1}\:+\frac{\mathrm{1}}{\infty}\:=\:\mathrm{1}\:+\:\mathrm{0}\:=\:\mathrm{1} \\ $$$$ \\ $$

Commented by alephzero last updated on 07/Feb/22

$$\mathrm{Oh},\:\mathrm{sorry}.\:\mathrm{I}\:\mathrm{forgot}\:\mathrm{about}\:\mathrm{that} \\ $$

Commented by TheSupreme last updated on 07/Feb/22

e^(ln(f(x))) =f(x) lim e^(ln(f(x))) =lim f(x)

$${e}^{{ln}\left({f}\left({x}\right)\right)} ={f}\left({x}\right) \\ $$$${lim}\:{e}^{{ln}\left({f}\left({x}\right)\right)} ={lim}\:{f}\left({x}\right) \\ $$$$ \\ $$

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Question-165698

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