Calculate the limit as $x\to0$ 
I need to calculate the limits as $x$ tends to $0$
For the first one, I get that the limit is zero, by splitting it up into $x^3(\sin(1/x))$ and $x^3(\sin^2(x))$ and using the sandwich theorem on $x^3(\sin(1/x))$ and just the algebra of limits for $x^3(\sin^2(x))$.
Is this correct?
For the second one, I am confused. I assume the limit should not exist, however when I try to use the sandwich theorem I keep getting that the limit is $-1$ and I don't think this is correct.
Could someone please help me.
Thanks 
 A: The first one you are correct.
The second one does not exists; in fact
$$\lim_{x \to 0^+} \frac{\sin(|x|)}{x^2 + \sin(x)} = \lim_{x \to 0^+}\frac{\sin(x)}{x^2 + \sin(x)} = 1$$
$$\lim_{x \to 0^-} \frac{\sin(|x|)}{x^2 + \sin(x)} =\lim_{x \to 0^-} \frac{-\sin(x)}{x^2 + \sin(x)} = -1$$
You're probably using the sandwich therem wrong :)
A: Another approach to show that the second limit does not exist is to use "asymptotics."  To that end, we have 
$$\sin|x|=|x|+O(|x|^3) \tag1$$ 
and 
$$x^2+\sin x =x+O(x^2). \tag 2$$  
Using $(1)$ and $(2)$, we find that
$$\frac{\sin|x|}{x^2+\sin x }=\frac{|x|+O(|x|^3)}{x+O(x^2)}=\frac{|x|}{x}+O(|x|)$$
which has a right-sided limit of $+1$ and a left-sided limit of $-1$.
A: Hint for the first one: 
If $f$ is bounded,and $g→0$ for $x→0$ , then $lim_{x→0}fg=0$
A: Your first result is correct -- the limit is indeed $0$ (you can also directly observe that the parenthesis takes values bounded in $[-2,2]$, and apply the sandwich theorem from there).
For the second, the limit does not exist. You can see it by recalling that $\lim_{y\to 0}\frac{\sin y}{y} =1$, and rewriting, for $x\neq 0$ sufficiently small:
$$
\frac{\sin \lvert x\rvert}{x^2 + \sin x} = \frac{\sin \lvert x\rvert}{\sin x }\frac{1}{\frac{x^2}{\sin x} + 1} = \frac{\lvert x\rvert}{x}\cdot\frac{\sin \lvert x\rvert}{\lvert x\rvert }\cdot\frac{x}{\sin x}\cdot\frac{1}{\frac{x}{\sin x}\cdot x + 1}
$$
which tends to either $-1$ or $1$ depending on whether $x\to 0$ from the right ($x>0$ or left $x<0$) -- each factor tends to $1$, except for the first that goes to $\pm 1$.
