Math Help - limit tan[x]/x

1. limit tan[x]/x

limit tan[x]/x
x->infinite

2. Re: limit tan[x]/x

Since $\tan x$ is surjective onto the set of reals, given any $k \in \mathbb{R}$, in each interval $\left(\dfrac{n\pi}{2},\dfrac{(n+1)\pi}{2}\right)$ , there exists $a_n$ such that $\tan(a_n) = ka_n$.

$\displaystyle \lim_{n \to \infty} a_n = \infty$

So, since $\displaystyle \lim_{n \to \infty} \dfrac{\tan(a_n)}{a_n} = k$ you know $\displaystyle \lim_{x \to \infty} \dfrac{\tan x}{x}$ does not exist.

Edit: made the argument more general.

3. Re: limit tan[x]/x

f(x)=sinx/xcosx
sinx=1 and xcosx=0 for x=Π/2+2nΠ, n=0,1,2,3,..
So there is no X st:
L-ε < f(x) < L+ε for all x>X

4. Re: limit tan[x]/x

؟؟؟؟؟؟؟؟؟
Because I thought the two before it came into my mind, but they are wrong

6. Re: limit tan[x]/x

The solution I posted is the correct answer to the problem you posted. I even proved it. If you think it is wrong, then you posted the wrong question. Is the correct question supposed to be $\displaystyle \lim_{x \to 0} \dfrac{\tan x}{x}$? Because that is a very different question. You asked the limit as $x \to \infty$, which does not exist.

7. Re: limit tan[x]/x

Originally Posted by alirezayi9094

limit tan[x]/x
x->infinite
The limit is undefined.

You can't talk about a limit in this case. To say the limit doesn’t exist is ambiguous. It could mean divergence, which is not the case here.

8. Re: limit tan[x]/x

is ur x greatest integer fn here

9. Re: limit tan[x]/x

Originally Posted by prasum
is ur x greatest integer fn here
Even if it is the greatest integer function, the limit would still be undefined. However, if it is the tangent of the fractional part of x over x, then that is different. Using the squeeze theorem, you have $0 \le \dfrac{\tan \langle x \rangle}{x} < \dfrac{\tan 1}{x}$

The limit as $x \to \infty$ of the LHS and RHS of the inequality are both 0, so by the Squeeze Theorem, so is the middle limit.

10. Re: limit tan[x]/x

Originally Posted by SlipEternal
$\dfrac{\tan \langle x \rangle}{x} < \dfrac{\tan 1}{x}$
That inequality is not true

11. Re: limit tan[x]/x

Originally Posted by Shakarri
That inequality is not true
Yes it is... as I said, if $\langle x \rangle$ means the fractional part of $x$, that is exactly what it means.