Intuitive understanding of limit cos(x)/x as x->0

by Moogie
Tags: cosx or x, intuitive, limit, x>0
 Emeritus Sci Advisor PF Gold P: 4,500 Here's a kind of brute force method: If $$\frac{-1}{2}2, then [tex]cos(x)<\frac{1}{2}$$ (since $$cos(\frac{\pi}{6}=\frac{1}{2}$$. All you really are interested in here is that cos(x) is bounded away from zero). Then if $$0  Emeritus Sci Advisor HW Helper Thanks PF Gold P: 11,722 If you look at the formal definition of the limit of a function, you'll see that for the limit to exist, (cos x)/x has to approach some real number L as x approaches 0. The fact that (cos x)/x is unbounded as you approach 0 from either side is enough to say that the limit doesn't exist. So when you write something like [tex]\lim_{x \to c} f(x) = \infty$$ you're actually saying the limit doesn't exist and you're saying why, i.e. f(x) blows up there. In comparison, if you look at the function f(x)=|x|/x as x approaches 0, then the one-sided limits do exist, so there's still the possibility that the two-sided limit exists. But in this case, the one-sided limits $$\lim_{x \to 0^+} \frac{|x|}{x} = 1$$ and $$\lim_{x \to 0^-} \frac{|x|}{x} = -1$$ aren't equal, so you would conclude from that that the limit doesn't exist.
 Emeritus Sci Advisor HW Helper Thanks PF Gold P: 11,722 No, the function isn't bounded from below when $x \to 0^-$, so it's unbounded on that side too. Your reasoning about how (cos x)/x behaves near x=0 is fine.