$\lim_{M\to\infty} \arctan M$

Question

$\lim_{M\to\infty} \arctan(M)$

I was trying:

$$ \lim_{M\to\infty} \arctan(M)$$

intuitively $\arctan(M)$ takes as input a number and outputs an angle the only point where I know the tangent line to the unit circle is infinity is $\frac{\pi}{2}$ then

$$ \lim_{M\to\infty} \arctan(M) = \frac{\pi}{2}$$

but is there any other way to prove thislimit without using intuition?

score 1 · Answer 1 · answered Jan 12 '24 at 16:10

1

Taking $M=\tan(x)$ yields $$\lim_{M\to \infty }\arctan(M)=\lim_{x\to \frac{\pi}{2}^-}\arctan(\tan(x))=\lim_{x\to \frac{\pi}{2}^-}x=\frac{\pi}{2}.$$

answered Jan 12 '24 at 16:10

Surb

55,662

I think your substitution needs some justification. What happens if someone chooses $M=\tan(x^2)$? – Bumblebee Jan 12 '24 at 16:17
is there other way that not uses the fact that $tan(pi/2) = \infty$? – samsamradas Jan 12 '24 at 16:20

$\lim_{M\to\infty} \arctan M$

1 Answers1