# Limit with Taylor Series

## Homework Statement

$$\lim_{x \to 0}[\frac{\sin(\tan(x))-\tan(\sin(x))}{x^7}]$$

## Homework Equations

$$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!} + ...$$
$$\tan(x)=x+\frac{x^3}{3}+\frac{2x^5}{15}+\frac{17x^7}{215}+ ...$$

## The Attempt at a Solution

I have an idea of how to do this by replacing sin(tan(x)) with tan(x) - tan(x)^3/3! + tan(x)^5/5!, etc., and then replacing the tans and sines with their respective taylor series. But I'm supposed to be able to do this without a calculator and presumably without expanding polynomials to the seventh power, which seems a bit ridiculous.

Can someone point me in the right direction?

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vela
Staff Emeritus
For example, consider the $x^3$ term in the expansion of $\sin(\tan x)$. Contributions to it come from the combination of the linear term in the expansion of tan x and the x3 of sin x, or from the combination of the x3 term of tan x and the linear term of sin x. So the x^3 term will be
$$\frac{x^3}{3} - \frac{x^3}{3!} = \frac{x^3}{6}$$