# Thread: Limit with Taylor formula

1. ## Limit with Taylor formula

"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}
$

2. 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 $|x| < \frac {\pi}2$

3. 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.

4. 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}

5. thank you
one more question, did u do those computations by hand?

6. 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.

7. 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...

8. 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.

9. 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

10. 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.