Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

,$\displaystyle \lim_{z \rightarrow \infty} \arctan(z) = \frac{\pi}{2} $. One way to see this is to put $\displaystyle z = \frac{y}{x}$ and imagine $y$ and $x$ as the sides of a right triangle. Then as $x$ goes to zero, $\displaystyle \frac{y}{x}$ goes to infinity and $\theta$ where $\theta = \arctan(z)$, goes to $\displaystyle \frac{\pi}{2}$. I have two questions regarding this problem:

  1. Is the geometric proof above, sufficiently rigorous for higher analysis? If not, why?

  2. Is there an analytic way to prove this i.e without any geometric intuition?

share|cite|improve this question
And also, sorry for the delay in replying. I'd not logged in for a couple of days. – Nikhil Panikkar Mar 3 '13 at 6:36
So sorry, it was my fault. I lookedat your answers and I thought it were your questions. So sorry. I'll remove my comment. – Git Gud Mar 3 '13 at 11:29
Its ok. I'll also delete my comment. :-) – Nikhil Panikkar Mar 4 '13 at 3:38
up vote 2 down vote accepted

First we prove the relation $$\forall z>0,\quad \arctan(z)+\arctan(\frac{1}{z})=\frac{\pi}{2}.$$ Indeed, let denote $f(z)=\arctan(z)+\arctan(\frac{1}{z})$ for $z>0$, then we check easily that $f'(z)=0$(use the fact that the derivative of $\arctan(z)$ is$\frac{1}{1+z^2}$), then $f$ must be a constant on the interval $(0,+\infty)$ and for $z=1$, since $\arctan(1)=\frac{\pi}{4}$ we find the result.

Now, it's easy to see that $$\lim_{z\rightarrow +\infty}\arctan z=\lim_{z\rightarrow +\infty}\frac{\pi}{2}-\arctan(\frac{1}{z})=\frac{\pi}{2}-\arctan 0=\frac{\pi}{2}.$$

To answer the first question I say, the geometric proof help to guess the result but it is not considered analytical proof.

share|cite|improve this answer
Hmm..., nice proof. Can you explain the answer to the first question in more detail? – Nikhil Panikkar Feb 28 '13 at 10:42
I mean, I am an engineering student, and I need to know when its ok, to use geometric intuition and when its not. – Nikhil Panikkar Feb 28 '13 at 10:49
@NikhilPanikkar In general the geometric intuition is very helpful mainly you are future engineer, but in my opinion it is very limited compared to analytical methods. – user63181 Feb 28 '13 at 11:36

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.