Find limit using Maclaurin series (remember the importance of big O notation)

Question

I have a problem that sounds like this:

Find the limit $$\lim_{x\rightarrow 0} \frac{14\tan(6x)-84x}{6x^3}$$ using Maclaurin series, and don't forget the importance of big O notation.

I have tried to find the Maclaurin series in different ways, but I always end up with the wrong answer. And I don't know how to use the big O notation in a helpful way here.

Thank you in advance.

Answer 1

Using the Maclaurin series of $\tan x$ and letting $x\leftarrow 6x$ yields

\begin{align} \lim_{x\rightarrow 0} \frac{14\tan(6x)-84x}{6x^3}&=\lim_{x\rightarrow 0} \frac{14\left \{ 6x+72x^3+O(x^5) \right \}-84x}{6x^3}\\ &=\lim_{x\rightarrow 0} \frac{84x+1008x^3+O(x^5)-84x}{6x^3}\\ &=168 \end{align}