Limit with Taylor formula

**xkyve** · November 9th 2008, 10:59 AM

"Calculate the limit using Taylor's formula". I know that the limit should be 1/30. Problem is how to prove it.

$ \lim_{x\rightarrow0}\frac{tan(sinx) - sin(tanx)}{x^7} $

**Jhevon** · November 9th 2008, 11:10 AM

Originally Posted by xkyve

"Calculate the limit using Taylor's formula". I know that the limit should be 1/30. Problem is how to prove it.

$ \lim_{x\rightarrow0}\frac{tan(sinx) - sin(tanx)}{x^7} $

hint:

$\sin x = x - \frac {x^3}{3!} + \frac {x^5}{5!} - + \cdots$ for all $x$

and

$\tan x = x + \frac {x^3}{3} + \frac {2x^5}{15} + \cdots$ for $|x| < \frac {\pi}2$

**xkyve** · November 9th 2008, 11:40 AM

i know the expansions and i found the value 17/315 + 1/7! for the limit but it's incorrect. i think i must find a path to write those trig. functions in another way.

**Mathstud28** · November 9th 2008, 11:58 AM

Let $f(x)=\tan(\sin(x))$ and $g(x)=\sin(\tan(x))$ . Then

$f(0)=0$ ---- $g(0)=0$
$f'(0)=1$ ---- $g'(0)=1$
$f''(0)=0$ ---- $g''(0)=0$
$f^{(3)}(0)=1$ ---- $g^{(3)}(0)=1$
$f^{(4)}(0)=0$ ---- $g^{(4)}(0)=0$
$f^{(5)}(0)=-3$ ---- $g^{(5)}(0)=-3$
$f^{(6)}(0)=0$ ---- $g^{(6)}(0)=0$
$f^{(7)}(0)=-107$ ---- $g^{(7)}(0)=-275$

$\therefore\quad\tan(\sin(x))=x+\frac{x^3}{6}-\frac{x^5}{40}-\frac{107x^7}{5040}\pm\cdots$

and $\sin(\tan(x))=x+\frac{x^3}{6}-\frac{x^5}{4}-\frac{275x^7}{5040}\pm\cdots$

$\begin{aligned}\therefore\quad&\lim_{x\to{0}}\frac {\tan(\sin(x))-\sin(\tan(x))}{x^7}\\ &=\lim_{x\to{0}}\frac{x+\frac{x^3}{6}-\frac{x^5}{40}-\frac{107x^7}{5040}\pm\cdots-\left(x+\frac{x^3}{6}-\frac{x^5}{4}-\frac{275x^7}{5040}\pm\cdots\right)}{x^7}\\ &=\lim_{x\to{0}}\frac{\frac{x^7}{30}\pm\cdots}{x^7 }\\ &=\frac{1}{30} \end{aligned}$

**xkyve** · November 10th 2008, 01:37 AM

thank you

one more question, did u do those computations by hand?

**Mathstud28** · November 10th 2008, 03:11 AM

Originally Posted by xkyve

thank you

one more question, did u do those computations by hand?

Up to the third derivative. And then I was like screw this and used my calculator.

**xkyve** · November 10th 2008, 04:06 AM

yeah, those computations are long... anyways, i was wondering if there was another method, one that envolves only a pencil and some paper.

i'll have to think about it...

**Mathstud28** · November 10th 2008, 12:15 PM

Originally Posted by xkyve

yeah, those computations are long... anyways, i was wondering if there was another method, one that envolves only a pencil and some paper.

i'll have to think about it...

There are other methods, but none that are using the Taylor method. Because that is the Taylor method.

**xkyve** · November 11th 2008, 02:58 AM

probably you are right, but i will still try to solve it using taylor but less computations.
doesn't matter, can u recommend me some good software that is able to compute derivatives, limits, and so on?

thanks

**Mathstud28** · November 11th 2008, 04:03 AM

Originally Posted by xkyve

probably you are right, but i will still try to solve it using taylor but less computations.
doesn't matter, can u recommend me some good software that is able to compute derivatives, limits, and so on?

thanks

Mathcad, TI-89, or Mathematica are all good choices.

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