Retaining the three terms in the series, estimate the remaining series using "Big Oh" notation with the best integer value possible, as $x\to 0$. The series is
$$\ln (\tan (x)) =\ln(x)+ \frac{x^2}{3} + \frac{7x^4}{90} + \frac{62x^6}{2835}+..., \quad \left( 0< \left|x\right| <\frac{\pi}{2} \right)$$
I was thinking of proving
$$f(x)=\left| \ln (\tan (x)) - \left(\ln(x)+ \frac{x^2}{3} + \frac{7x^4}{90} \right) \right| \leq C \left|x^4\right|$$
where $g(x)=x^4$.
Then i'd calculate this
$$ \left| \frac{f(x)}{g(x)} \right| = \left| \frac{\ln (\tan (x)) - \left( \ln(x)+ \frac{x^2}{3} + \frac{7x^4}{90} \right)} {x^4} \right|$$
I think the easiest way would be using this method where you choose $k=1$ and $x>1$, but i'm not sure how to do it with this expression as there are fractions, the natural logarithm and even tangens.
The problem, here b):
