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### Evaluating $\lim_{x\to0}\frac{x\sin{(\sin{x})}-\sin^2{x}}{x^6}$ [duplicate]

Evaluate: $$\lim_{x\to0}\frac{x\sin{(\sin{x})}-\sin^2{x}}{x^6}$$ I have been trying to solve this for $15$ minutes but sin(sin(x)) part has me stuck. My attempt:...
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### Solve the following limit as $\lim_{x \to 0}$ [duplicate]

$$\lim_{x \to 0} \frac{x\sin(\sin x) - \sin^2 x}{x^6}$$ **My Attempt: ** I started with L'Hopital's rule. But it quickly became messy. So, I did not continue. I tried to write the Taylor series of ...
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### How to calculate $\lim_{x\to 0}\left(\frac{1}{x^2} - \frac{1}{\sin^2 x}\right)^{-1}$? [closed]

$$f (x) = \frac{1}{x^2} - \frac{1}{\sin^2 x}$$ Find limit of $\dfrac1{f(x)}$ as $x\to0$.
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### How to compute $\lim_{x\to 0^+}\frac{\arctan x-x}{x^2}$ without Taylor's formula or L'Hôpital's rule?

I have to find $$\lim_{x \rightarrow 0^+} \frac{ \arctan(x)-x}{x^2}$$ without Taylor's formula or L'Hôpital's rule. How to tackle it? Any idea is welcome.
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### Examples of limits that become easier with Taylor series

There are examples of questions on this site where the OP asks for help solving a limit problem, and some of the answers make use of clever Taylor expansions to evaluate the limit. The purpose of this ...
### Evaluating $\lim_{x\to 0}\frac{\sin(x)\arcsin(x)-x^2}{x^6}$ Step by Step Using L' Hopital Rule
The limit to be found is $$\lim_{x\to 0}\frac{\sin(x)\arcsin(x)-x^2}{x^6}$$ I've tried l'hopital rule but it gets really messy. I've also tried splitting it into 2 limits but that doesn't work. I ...
### How to calculate $\lim \limits_{x \to 0} \frac{x^2 \sin^2x}{x^2-\sin^2x}$ with $\lim \limits_{x \to 0} \frac{\sin x}{x}=1$?
How to calculate $$\lim \limits_{x \to 0} \frac{x^2 \sin^2x}{x^2-\sin^2x}$$ with $$\lim \limits_{x \to 0} \frac{\sin x}{x}=1?$$ Yes I know the question has been asked, the answer is $3$, L'Hospital ...