# question on arcsec infinity

#### FailCalculus

can someone explain to me how arcsec infinity is equal to pi/2? thanks

can someone explain to me how arcsec infinity is equal to pi/2? thanks
$$\displaystyle \sec \theta=\frac{1}{\cos \theta}$$

Set the denominator to be 0

#### HallsofIvy

MHF Helper
can someone explain to me how arcsec infinity is equal to pi/2? thanks
Strictly speaking, "$$\displaystyle arcsec(\infty)$$" isn't $$\displaystyle \pi/2$$ or any other number. "$$\displaystyle \infty$$" isn't a number so secant is not defined for it. You could, of course, extend the real number system to include "infinities" but there are several different ways to do that- you would have to specify which you meant.

But, we can talk about this in the limit sense. $$\displaystyle sec(\theta)= \frac{1}{cos(\theta)}$$ and as $$\displaystyle \theta$$ goes to $$\displaystyle \pi/2$$, $$\displaystyle cos(\theta)$$ goes to 0 and so $$\displaystyle sec(\theta)= \frac{1}{cos(\theta)$$ "goes to infinity" (get larger without bound). Thus, if we extend the real numbers sytem in that way, we have that "$$\displaystyle sec(\pi/2)= \infty$$" and so "$$\displaystyle arcsec(\infty)= \pi/2$$".

• FailCalculus