<s>[INST] Verify if $e^{x^{3}} - \sqrt[4]{1-x^{4}} + x^2 \sin(x)$ is a little-o of $x^{2}$ as $x \to 0^{+}$ using limit I have to verify if $e^{x^{3}} - \sqrt[4]{1-x^{4}} + x \sin(x)$ is a little-o of $x^{2}$ as $x \to 0^{+}$ using limit, i.e. calculating the limit
$$\lim_{x \to 0^{+}} \frac{e^{x^{3}} - \sqrt[4]{1-x^{4}} + x^2 \sin(x)}{x^{2}} $$
and if the result is $0$, then the answer is "yes".
My first question is: is it acceptable that $x \to 0^{+}$? I mean: I always did exercises with limit from left and right. Is the little-o notation defined also with limit only from right side or left side?
My second question is: ho can I calculate the limit?
Thanks
EDIT
My attempt of solution using L'Hopital's rule.
$$\lim_{x \to 0^{+}} \frac{e^{x^{3}} - \sqrt[4]{1-x^{4}} + x^2 \sin(x)}{x^{2}} \\
\lim_{x \to 0^{+}} \frac{e^{x^{3}} - \sqrt[4]{1-x^{4}}}{x^{2}}+\lim_{x \to 0^{+}} \frac{x^{2}\sin(x)}{x^{2}}\\
\lim_{x \to 0^{+}} \frac{\frac{d}{dx}(e^{x^{3}}-\sqrt[4]{1-x^{4}})}{\frac{d}{dx}x^{2}} + \lim_{x \to 0^{+}}\sin(x)\\
\lim_{x \to 0^{+}} \frac{x^{2}\left(\frac{x}{\sqrt[4]{(1-x^{4})^{3}}}+3e^{x^{3}}\right)}{2x} + 0\\
\lim_{x \to 0^{+}}\frac{x}{2}\left(\frac{x}{\sqrt[4]{(1-x^{4})^{3}}}+3e^{x^{3}}\right)\\
\frac{0}{2}\left(\frac{0}{1}+3\right)=
0
$$
Is it OK?
EDIT #2
My attempt of solution following the hint by Claude Leibovici
$$\require{cancel}\lim_{x \to 0^{+}} \frac{e^{x^{3}} - \sqrt[4]{1-x^{4}} + x^2 \sin(x)}{x^{2}} \\
\lim_{x \to 0^{+}} \frac{1+x^{3}-(1-x^{4})+x^{2}x}{x^{2}}\\
\lim_{x \to 0^{+}}\frac{1+x^3-1+x^4+x^3}{x^2}\\
\lim_{x \to 0^{+}}\frac{x^4+2x^3}{x^2}\\
\lim_{x \to 0^{+}}\frac{\cancel{x^{2}}(x^{2}+2x)}{\cancel{x^{2}}}\\
\lim_{x \to 0^{+}} x^2+2x = 0 
$$
Is it OK?
 [/INST] We can simply using asymptotics relations:
$$\lim_{x \to 0} \frac{e^{x^{3}} - \sqrt[4]{1-x^{4}} + x^2 \sin(x)}{x^{2}}\,\,=\,\,\lim_{x \to 0} \frac{\left(e^{x^{3}} - 1\right)+\left(1-\sqrt[4]{1-x^{4}}\right) + x^2 \sin(x)}{x^{2}}\,\,\sim\,\,\lim_{x \to 0^{+}} \frac{x^3+\frac{1}{4}x^4+x^3}{x^2}=\lim_{x\to 0}\frac{o(x^2)}{x^2}=0$$
So, $e^{x^{3}} - \sqrt[4]{1-x^{4}} + x^2 \sin(x) = o(x^2)$ as you wanted to show.
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