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Question Number 75694 by malwaan last updated on 15/Dec/19

prove that lim_(x→∞) x^(1/x) =1

$$\boldsymbol{{prove}}\:\boldsymbol{{that}}\: \\ $$$$\underset{\boldsymbol{{x}}\rightarrow\infty} {\boldsymbol{{lim}}}\:\boldsymbol{{x}}^{\frac{\mathrm{1}}{\boldsymbol{{x}}}} \:=\mathrm{1} \\ $$

Answered by vishalbhardwaj last updated on 15/Dec/19

(as x→∞ then (1/x) → 0) ⇒ lim_(x→ ∞) x^(lim_(x→∞) (1/x)) = lim_(x→∞) (x)^0 = 1

$$\left({as}\:{x}\rightarrow\infty\:\mathrm{then}\:\frac{\mathrm{1}}{{x}}\:\rightarrow\:\mathrm{0}\right) \\ $$$$\Rightarrow\:{li}\underset{{x}\rightarrow\:\infty} {{m}x}^{{li}\underset{{x}\rightarrow\infty} {{m}}\frac{\mathrm{1}}{{x}}} =\:{li}\underset{{x}\rightarrow\infty} {{m}}\left({x}\right)^{\mathrm{0}} \:=\:\mathrm{1} \\ $$

Commented by malwaan last updated on 15/Dec/19

but lim_(x→∞) x^0 = ∞^0 undefind

$${but}\:\underset{{x}\rightarrow\infty} {{lim}}\:{x}^{\mathrm{0}} \:=\:\infty^{\mathrm{0}} \:{undefind} \\ $$

Commented by vishalbhardwaj last updated on 15/Dec/19

whatever the value if x is there, but the whole power is zero, that why its value will be zero in the sense

$$\mathrm{whatever}\:\mathrm{the}\:\mathrm{value}\:\mathrm{if}\:{x}\:\mathrm{is}\:\mathrm{there},\:\mathrm{but} \\ $$$$\mathrm{the}\:\mathrm{whole}\:\mathrm{power}\:\mathrm{is}\:\mathrm{zero},\:\mathrm{that}\:\mathrm{why} \\ $$$$\mathrm{its}\:\mathrm{value}\:\mathrm{will}\:\mathrm{be}\:\mathrm{zero}\:\mathrm{in}\:\mathrm{the}\:\mathrm{sense} \\ $$

Commented by MJS last updated on 15/Dec/19

you cannot split it like this lim_(x→∞) x^(1/x) ≠(lim_(x→∞) x)^((lim_(x→∞) (1/x))) because lim_(x→∞) x =∞∉R and r^0 =1 only for r∈R\{0}

$$\mathrm{you}\:\mathrm{cannot}\:\mathrm{split}\:\mathrm{it}\:\mathrm{like}\:\mathrm{this} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}{x}^{\frac{\mathrm{1}}{{x}}} \neq\left(\underset{{x}\rightarrow\infty} {\mathrm{lim}}{x}\right)^{\left(\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{{x}}\right)} \\ $$$$\mathrm{because}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}{x}\:=\infty\notin\mathbb{R} \\ $$$$\mathrm{and}\:{r}^{\mathrm{0}} =\mathrm{1}\:\mathrm{only}\:\mathrm{for}\:{r}\in\mathbb{R}\backslash\left\{\mathrm{0}\right\} \\ $$$$ \\ $$

Answered by MJS last updated on 15/Dec/19

lim_(x→∞) x^(1/x) =lim_(x→∞) e^((ln x)/x) =e^(lim_(x→∞) ((ln x)/x)) lim_(x→∞) ((ln x)/x)=lim_(x→∞) (((d/dx)[ln x])/((d/dx)[x]))=lim_(x→∞) (1/x)=0 ⇒ e^(lim_(x→∞) ((ln x)/x)) =1 ⇒ lim_(x→∞) e^((ln x)/x) =1 ⇒ lim_(x→∞) x^(1/x) =1

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}{x}^{\frac{\mathrm{1}}{{x}}} =\underset{{x}\rightarrow\infty} {\mathrm{lim}e}^{\frac{\mathrm{ln}\:{x}}{{x}}} =\mathrm{e}^{\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{ln}\:{x}}{{x}}} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{ln}\:{x}}{{x}}=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\frac{{d}}{{dx}}\left[\mathrm{ln}\:{x}\right]}{\frac{{d}}{{dx}}\left[{x}\right]}=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{{x}}=\mathrm{0} \\ $$$$\Rightarrow\:\mathrm{e}^{\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{ln}\:{x}}{{x}}} =\mathrm{1}\:\Rightarrow\:\underset{{x}\rightarrow\infty} {\mathrm{lim}e}^{\frac{\mathrm{ln}\:{x}}{{x}}} =\mathrm{1}\:\Rightarrow\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}{x}^{\frac{\mathrm{1}}{{x}}} =\mathrm{1} \\ $$

Commented by vishalbhardwaj last updated on 15/Dec/19

sir, Is my assumption and explantion wrong ??

$$\mathrm{sir},\:\mathrm{Is}\:\mathrm{my}\:\mathrm{assumption}\:\mathrm{and}\:\mathrm{explantion}\:\mathrm{wrong}\:?? \\ $$

Commented by malwaan last updated on 16/Dec/19

thank you so much

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

Answered by $@ty@m123 last updated on 15/Dec/19

Solution: Let y= lim_(x→∞) x^(1/x) ln y= lim_(x→∞) (1/x)ln x ⇒ ln y= 0 ⇒ y=1

$${Solution}: \\ $$$${Let}\:\:{y}=\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\:{x}^{\frac{\mathrm{1}}{{x}}} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{ln}\:{y}=\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\:\:\frac{\mathrm{1}}{{x}}\mathrm{ln}\:{x} \\ $$$$\:\Rightarrow\:\:\:\mathrm{ln}\:{y}=\:\mathrm{0} \\ $$$$\:\Rightarrow\:{y}=\mathrm{1}\:\:\:\:\:\:\: \\ $$

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