# Epsilon - delta proof: $\lim\limits_{x\to\frac\pi2} \tan x$ does not exist.

I am required to prove that the $$\lim\limits_{x\to\frac\pi2} \tan x=\text{DNE}$$ using the epsilon-delta definition. I am trying to do it by contradiction, but I got stuck.

Let us assume that $$\lim\limits_{x \to \frac\pi2} \tan x = L$$

This means that

$$\forall \epsilon>0,\; \exists \delta>0$$ such that $$0 < |x -\frac\pi2| < \delta \implies |\tan x - L| < \epsilon$$

My approach is to propose an $$\epsilon$$, say, $$\frac{1}{4}$$ and then using some value of $$x$$ showing that the difference between $$\tan x$$ and $$L$$ is larger than the $$\epsilon$$.

Should I consider cases? For example:

$$L = 0$$, $$L>0$$, $$L < 0$$

I have also thought to use $$\arctan$$ to choose an appropriate $$x$$ because I understand $$|x-\frac\pi2|$$ must be less than $$\delta$$.

Any suggestion on how to proceed?

• The limit from the right is $\infty$. The limit from the left is $-\infty$. Oct 20, 2019 at 19:04

Forget proof by contradiction. It's a lot simpler to simply compute the directional limits and compare them.

Note that $$\lim\limits_{x\to\frac\pi2^+} \tan x=-\infty,\lim\limits_{x\to\frac\pi2^-} \tan x=\infty$$

Note that $$\tan(x)$$ is odd about $$x=\pi/2$$. So, it might be easier to proceed by making the simple change of variable $$x\mapsto x+\pi/2$$.

Hence, it suffices to show that the limit $$\lim_{x\to 0}\cot(x)$$ fails to exist. In that which follows, we shall invoke the inequality $$x\le \sin(x)$$, for $$x<0$$.

First we require that $$-\pi/3. Then, for any number $$L>0$$, however large, we have

$$\cot(x)<\frac{1}{2\sin(x)}<\frac1{2x}\le -L$$

whenever $$x\ge -\frac1{2L}$$. So, we find that for any $$L>0$$,

$$\cot(x) <-L$$

whenever $$\max\left(-\frac\pi3,-\frac1{2L}\right). This is equivalent to the statement $$\lim_{x\to 0^-}\cot(x)=-\infty$$.

Proceeding analogously for $$x>0$$, we find that $$\lim_{x\to 0^+}\cot(x)=+\infty$$.

Inasmuch as the limit from the right and left are unequal, the limit of interest fails to exist. And we are done.

• I see that when you make the substitution by the sum of angles for sine and cosine you get cot(x) but I fail to understand why it is valid to change the limit to cot(x). Could you help me to understand why? Oct 20, 2019 at 20:59
• Please feel free to up vote an answer as you see fit Nov 11, 2019 at 23:26