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Question Number 15186 by Joel577 last updated on 08/Jun/17

lim_(x→∞)  (((x + 3)/(x −1)))^x

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{{x}\:+\:\mathrm{3}}{{x}\:−\mathrm{1}}\right)^{{x}} \\ $$

Commented by mrW1 last updated on 08/Jun/17

yes, correct.

$$\mathrm{yes},\:\mathrm{correct}. \\ $$

Commented by Joel577 last updated on 08/Jun/17

thank you very much

$${thank}\:{you}\:{very}\:{much} \\ $$

Commented by Joel577 last updated on 08/Jun/17

= lim_(x→∞)  (((x − 1 + 4)/(x − 1)))^x   = lim_(x→∞)  (1 + (4/(x − 1)))^x   = lim_(x→∞)  (1 + (4/(x − 1)))^((x − 1) + 1)   = lim_(x→∞)  (1 + (4/(x − 1)))^(x − 1)  . lim_(x→∞)  (1 + (4/(x − 1)))  = e^4  . 1 = e^4     Is it correct?

$$=\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{{x}\:−\:\mathrm{1}\:+\:\mathrm{4}}{{x}\:−\:\mathrm{1}}\right)^{{x}} \\ $$$$=\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{1}\:+\:\frac{\mathrm{4}}{{x}\:−\:\mathrm{1}}\right)^{{x}} \\ $$$$=\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{1}\:+\:\frac{\mathrm{4}}{{x}\:−\:\mathrm{1}}\right)^{\left({x}\:−\:\mathrm{1}\right)\:+\:\mathrm{1}} \\ $$$$=\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{1}\:+\:\frac{\mathrm{4}}{{x}\:−\:\mathrm{1}}\right)^{{x}\:−\:\mathrm{1}} \:.\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{1}\:+\:\frac{\mathrm{4}}{{x}\:−\:\mathrm{1}}\right) \\ $$$$=\:{e}^{\mathrm{4}} \:.\:\mathrm{1}\:=\:{e}^{\mathrm{4}} \\ $$$$ \\ $$$$\mathrm{Is}\:\mathrm{it}\:\mathrm{correct}? \\ $$

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