x 0^+lim (5(x)^(1)/(3)) _e(1+3x^2)( ^-13x)^2(e^5(x)^(4)/(3)-1) is equal to

Question

$\underset{x	o {0}^{+}}{lim}\frac{	an (5(x{)}^{\frac{1}{3}}){\log }_{e}(1+3{x}^{2})}{{({	an }^{-1}3\sqrt{x})}^{2}({e}^{5(x{)}^{\frac{4}{3}}}-1)}$ is equal to

Accepted Answer

$\lim _{x \rightarrow 0^{+}} \frac{\tan \left(5(x)^{\frac{1}{3}}\right) \ln \left(1+3 x^3\right)}{\left(\tan ^{-1}(3 \sqrt{x})\right)^2\left(e^{5 x^{\frac{4}{3}}}-1\right)}$

$$\mathop {\lim }\limits_{x \to {0^ + }} {{{{\tan \left( {5{{(x)}^{{1 \over 3}}}} \right)} \over {5{{(x)}^{{1 \over 3}}}}}{{\ln (1 + 3{x^2})} \over {3{x^2}}} \times 5{{(x)}^{{1 \over 3}}}(3{x^2})} \over {{{{{\left( {{{\tan }^2}\left( {3\sqrt x } \right)} \right)}^2}} \over {{{\left( {3\sqrt x } \right)}^2}}}{{\left( {{e^{5x{4 \over 3}}} - 1} \right)} \over {5{x^{{1 \over 3}}}}} \times 9x \times 5{x^{{4 \over 3}}}}}$$

$$\mathop {\lim }\limits_{x \to {0^ + }} {{{{\tan \left( {5{{(x)}^{{1 \over 3}}}} \right)} \over {5{{(x)}^{{1 \over 3}}}}}{{\ln (1 + 3{x^3})} \over {3{x^2}}} \times 15{x^{{7 \over 3}}}} \over {{{{{\left( {{{\tan }^2}\left( {3\sqrt x } \right)} \right)}^2}} \over {{{\left( {3\sqrt x } \right)}^2}}}{{\left( {{e^{5(x){4 \over 3}}} - 1} \right)} \over {5{x^{{1 \over 3}}}}} \times 45{x^{{7 \over 3}}}}}$$

$$ = {1 \over 3}$$

Solutions · 1