Question: I've been stuck on this problem on my online math system centre, and I've tried everything. I have to send in work for the professor to mark, so I cannot graph it or use plot points. Additionally, I CANNOT use L'Hospital's rule, as it is not covered yet in course material. I apologize in advance, I'm new on StackExchange, but here is the problem:

$$\lim\_{x\to0}\dfrac{\sin4x}{x^2+x}$$

So far, I took the derivative of the equation and got

$$\frac{4\cos4x-2x^{-1}\sin4x-x^{-2}\sin(4x)}{x^{2}+x}$$

But since the limit is still zero, it will still be indeterminate. Since the denominator is $x^{2}+x$, it will always be some variant of that to the nth power. So it's become clear to me I cannot solve this with derivatives. I thought about using $$\lim\_{h\to0} \frac{f(x+h)-f(x)}h$$ but that didn't work out either. 

Can someone help me here? Regards, Joe

Answer: 
I've written some possible approaches.

**FIRST**

\begin{align\*}
\lim\_{x\rightarrow 0}\frac{\sin(4x)}{x^{2} + x} = \lim\_{x\rightarrow 0}\frac{\sin(4x)}{x(x+1)} = \lim\_{x\rightarrow 0}\frac{4\sin(4x)}{4x}\times\lim\_{x\rightarrow 0}\frac{1}{x+1} = \lim\_{y\rightarrow 0}\frac{4\sin(y)}{y} = 4
\end{align\*}

**SECOND**
\begin{align\*}
\lim\_{x\rightarrow 0}\frac{\sin(4x)}{x^{2} + x} & = \lim\_{x\rightarrow 0}\frac{\sin(4x) - \sin(0)}{4x - 0}\times\lim\_{x\rightarrow 0}\frac{4}{1+x} = 4\times\lim\_{y\rightarrow 0}\frac{\sin(y) - \sin(0)}{y - 0} = 4\cos(0) =4
\end{align\*}

**THIRD**

Since the Taylor series of the sine function is given by

\begin{align\*}
\sin(x) = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \ldots
\end{align\*}

We have $\sin(4x) \sim 4x + O(x^{3})$ when $x$ is near to zero. Hence we obtain:
\begin{align\*}
\lim\_{x\rightarrow 0}\frac{\sin(4x)}{x^{2}+x} = \lim\_{x\rightarrow 0}\frac{4x}{x^{2}+x} = \lim\_{x\rightarrow 0}\frac{4}{x+1} = 4
\end{align\*}

**FOURTH**

\begin{align\*}
\lim\_{x\rightarrow 0}\frac{\sin(4x)}{x^{2}+x} = \lim\_{x\rightarrow 0}\frac{2\sin(2x)\cos(2x)}{x(x+1)} = 4\times\lim\_{x\rightarrow 0}\frac{\sin(2x)}{2x}\times\lim\_{x\rightarrow 0}\frac{\cos(2x)}{x+1} = 4\times 1\times 1 = 4
\end{align\*}
Write your term as $$\frac{\sin(4x)}{4x}\cdot \frac{4}{x+1}$$