_x0^+(5x^1/3)(1+3x^2)(^-1(3 x))^2(e^5x^4/3-1) is equal to

Question

$\displaystyle\lim_{x	o0^+}\frac{	an\left(5x^{1/3}ight)\ln(1+3x^2)}{\left(	an^{-1}\left(3\sqrt xight)ight)^2\left(e^{5x^{4/3}}-1ight)}$ is equal to

Accepted Answer

Rewrite each factor using its own small-argument scale and apply the standard limits $\dfrac{\tan u}{u}\to1$, $\dfrac{\ln(1+v)}{v}\to1$, $\dfrac{\tan^{-1}w}{w}\to1$, $\dfrac{e^z-1}{z}\to1$:

$$\frac{\tan(5x^{1/3})}{5x^{1/3}}\cdot\frac{\ln(1+3x^2)}{3x^2}\cdot\frac{(3\sqrt x)^2}{(\tan^{-1}3\sqrt x)^2}\cdot\frac{5x^{4/3}}{e^{5x^{4/3}}-1}\cdot\frac{5x^{1/3}\cdot3x^2}{9x\cdot5x^{4/3}}$$

Each of the bracketed ratios tends to $1$, and the remaining algebraic factor of powers of $x$ reduces to a constant. Carrying out the power-counting (all exponents of $x$ cancel exactly, as the problem is constructed so the limit is finite) yields

$$\lim_{x\to0^+}(\cdots)=\frac13$$

Solutions · 1