how do you prove that the unction f(x) = tan(x+x^2)/(1+x+x^2) is continuous for all x=[0,1] except at one point.

i found out that it is not cont at x= 0.84936..but i dont know how to prove that it is continuous now..

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- Nov 17th 2010, 07:39 AMalexandrabel90continuous
how do you prove that the unction f(x) = tan(x+x^2)/(1+x+x^2) is continuous for all x=[0,1] except at one point.

i found out that it is not cont at x= 0.84936..but i dont know how to prove that it is continuous now.. - Nov 17th 2010, 08:53 AMchisigma
The function $\displaystyle f(\theta) = \tan \theta$ has a singularity for $\displaystyle \theta= \frac{\pi}{2}$ so that Your function has a singularity for x satisfying the equation $\displaystyle x^{2}+x-\frac{\pi}{2}=0$. One solution is $\displaystyle \displaystyle x= \frac{-1 + \sqrt{1+2 \pi}}{2} = .849368862...$

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$ - Nov 17th 2010, 08:59 AMalexandrabel90
haha ya i knew that it is not continuous at x= 0.84936.. as stated in my question..so, im trying to ask how do i show that it is continuous for all points other than x= 0.84936..

- Nov 17th 2010, 09:00 AMFernandoRevilla
The function:

$\displaystyle f(x)=\dfrac{\tan (x+x^2)}{1+x+x^2}$

is an elementary function, $\displaystyle 1+x+x^2\neq 0$ for all $\displaystyle x$ real and $\displaystyle \tan \pi/2$ does not exists. Solving $\displaystyle x^2+x=\pi/2$ you'll obtain the point in $\displaystyle [0,1]$ where $\displaystyle f$ is not continuous:

$\displaystyle x=\dfrac{-1+\;\sqrt[]{1+2\pi}}{2}\in{[0,1]}$

Regards.

Edited: Sorry, I didn't see the previous answers. - Nov 17th 2010, 09:09 AMFernandoRevilla
- Nov 17th 2010, 09:19 AMalexandrabel90
may i know how do i prove this theorem? i never learnt this theorem before. thanks!!

- Nov 17th 2010, 09:47 AMFernandoRevilla
For example:

**(i)**$\displaystyle f_1(x)=1+x+x^2$ is continuous in $\displaystyle [0,1]$ (polynomical function).

**(ii)**$\displaystyle f_2(x)=x+x^2$ is continuous in $\displaystyle [0,1]$ (polynomical function).

**(iii)**$\displaystyle f_3(x)=\tan x$ is continuous in $\displaystyle \ldots \cup (-\pi/2,\pi/2)\cup\ldots$

**(iv)**Composition of continuous are continuous.

etc. etc., ...

Regards.