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**Ackbeet** You'd like a sequence of inequalities that look like this:

$\displaystyle \dfrac{6x^2-7\sin(x)}{3x+1} >...>...>1000.$

So, if you can make the expression smaller but still with a size that you can control, you're good.

Since you're subtracting the sin function, and you want your function to get smaller, make the $\displaystyle 7\sin(x)$ term as big as possible:

$\displaystyle \dfrac{6x^2-7\sin(x)}{3x+1}\ge \dfrac{6x^2-7}{3x+1}.$

Now work with this, and make it bigger than the desired amount.