Shortcuts for Solving Brutal Limit Problems

[shakgoku]

Homework Statement

Limit x tends to zero, (sin(tan(x))-tan(sin(x)))/x^7

Homework Equations

The Attempt at a Solution

I have tried to derive a maclaurin expansion for sin(tan(x)) and tan(sin(x)).
But it was too lengthy to be attempted in an examination.
Another method is to use l'hospital but this method is also incredibly lengthy.

Is there a shorter method to attack this problem?

 

[lanedance]

you could attempt to use just the taylor series for tan & sin to expand it into some thing reasonable...

 

[lanedance]

expanding each to O(x^8) should be more than enough, as those and any higher terms will tend to zero in the limit

 

[shakgoku]

expanding each to O(x^8) should be more than enough, as those and any higher terms will tend to zero in the limit

Yeah, but calculating derivatives of sin(tan(x)) and tan(sin(x)) for taylor series, and keeping track of everything is a very long process.

I was wondering if an easier and shorter method is avaliable.

 

[lanedance]

i was saying only use the taylor series of sin(y) and tan(x)... the take the composition of polynomial functions

 

[shakgoku]

$$\sin\left(\tan\left(x\right)\right) = -\frac{1}{5040} \, tan(x)^{7} + \frac{1}{120} \, tan(x)^{5} - \frac{1}{6} \, tan(x)^{3} + tan(x)$$

and

$$\tan\left(\sin\left(x\right)\right) = \frac{17}{315} \, sin(x)^{7} + \frac{2}{15} \, sin(x)^{5} + \frac{1}{3} \, sin(x)^{3} +<br /> sin(x)$$
(x^8 is zero for both functions). Then ,
$$\frac{\sin\left(\tan\left(x\right)\right) -<br /> \tan\left(\sin\left(x\right)\right)}{x^{7}} =<br /> <br /> -\frac{272 \, \sin\left(x\right)^{7} + 672 \, \sin\left(x\right)^{5} +<br /> 1680 \, \sin\left(x\right)^{3} + \tan\left(x\right)^{7} - 42 \,<br /> \tan\left(x\right)^{5} + 840 \, \tan\left(x\right)^{3} + 5040 \,<br /> \sin\left(x\right) - 5040 \, \tan\left(x\right)}{5040 \, x^{7}}$$

and then how do I proceed?

 

[lanedance]

now substitute in the taylor expansions of sin & cos

it will seem messy at first, but as you only need keep terms up to O(x^7) you can throw away a lot

$$tan(x)^{7} = (\frac{17}{315}x^7 + \frac{2}{15}x^5 \ + \frac{1}{3} x^3 + x)^7 = x^7 + O(x^8)<br />$$