lim-x-0-x-1-ln-xsin-x-3-

Question Number 172872 by mnjuly1970 last updated on 02/Jul/22

\lim_{x \to 0^{+}} (x^{\frac{1}{l n (x s i n (x^{3}))}}) = ?

Answered by mnjuly1970 last updated on 02/Jul/22

e^{\lim_{x \to 0^{+}} \ln x^{\frac{1}{(\ln (xsin (x^{3}))}}} = e^{\lim_{x \to 0^{+}} \frac{\ln (x)}{\ln (x) + \ln (\sin (x^{3}))}}

= e^{\lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{1}{x} + \frac{{3 x}^{2} \cos (x^{3})}{\sin (x^{3})}}} =

= e^{\lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{1}{x} + \frac{3}{x}} = \lim (\frac{1}{4})} = \sqrt[4]{e}

Answered by Mathspace last updated on 02/Jul/22

= {l i m}_{x \to 0^{+}} e^{\frac{l n x}{x l n (s i n x)}}

{l i m}_{x \to 0^{+}} x l n (s i n x) = {l i m}_{x \to 0^{+}} x l n x = o^{-}

{l i m}_{x \to 0^{+}} l n x = - \infty \Rightarrow

{l i m}_{x \to 0^{+}} e^{\frac{l n x}{x l n (s i n x)}} = e^{+ \infty} = + \infty

Commented by mnjuly1970 last updated on 02/Jul/22

pls recheck the answer sir

Answered by floor(10²Eta[1]) last updated on 02/Jul/22

L = {\underset{x \to 0^{+}}{\lim x}}^{\frac{1}{\ln (xsin (x^{3}))}}

lnL = \lim_{x \to 0^{+}} \frac{lnx}{\ln (xsin (x^{3}))} = \lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{\sin (x^{3}) + {3 x}^{3} \cos (x^{3})}{xsin (x^{3})}}

= \lim_{x \to 0^{+}} \frac{\sin (x^{3})}{\sin (x^{3}) + {3 x}^{3} \cos (x^{3})}

= \lim_{x \to 0^{+}} \frac{{3 x}^{2} \cos (x^{3})}{{12 x}^{2} \cos (x^{3}) - {9 x}^{4} \sin (x^{3})}

= \lim_{x \to 0^{+}} \frac{3 \cos (x^{3})}{12 \cos (x^{3}) - {9 x}^{2} \sin (x^{3})} = \frac{3}{12} = \frac{1}{4}

\Rightarrow lnL = \frac{1}{4} \Rightarrow L = \sqrt[4]{e}

Commented by mnjuly1970 last updated on 03/Jul/22

thanks alot \dots very nice solution

Answered by a.lgnaoui last updated on 04/Jul/22

l n L = l n [x^{\frac{1}{l n (x \sin (x^{3}))}}] = \frac{1}{l n (x \sin (x^{3}))} \times l n x

= \frac{l n x}{l n (x \sin (x^{3}))} = \frac{l n x}{l n [x^{4} \times (\frac{\sin (x^{3})}{x^{3}})]}

{l i m}_{x \to 0} (\frac{\sin x}{x}) = {l i m}_{x \to 0} \frac{\sin (x^{3})}{x^{3}} = 1

l n L = \frac{l n x}{l n (x^{4})} = \frac{1}{4} \Rightarrow L = e^{\frac{1}{4}} =^{4} \sqrt{e} .