Compute the limit: _x 0 ( (( x) - ( x))/(x7) )

let's tackle these limit problems.

:

We need to compute the limit:

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

This limit is of the indeterminate form $\frac{0}{0}$, so we will use Taylor series expansions around $x=0$.

Step 1: Write down the Taylor series expansions for $\sin u$ and $\tan u$ up to the $x^7$ term.

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