Question

I am at loss of words: How does cot(x^2) get to be infinity when its approaching zero?

Answer 1

\[\frac{ \cos x ^{2}}{ sinx ^{2} }\]

Answer 2

the limit of cot(x) as it approaches 0 is infinity or undefined because cos(x)/sin(x) at 0 is cos(0)/sin(0) = 1/0

Answer 3

i thought as much, but how come we say its infinity because its zero or undefined as you said @bloopman

Answer 4

\[\lim_{x \rightarrow 0}\cot x^2 = \lim_{x \rightarrow 0}\frac{ \cos x^2 }{ \sin x^2 }=\lim_{x \rightarrow 0^+}\frac{ \cos x^2 }{ \sin x^2 }\]
this is because \(x^2 \ge 0\)

Answer 5

As x tends to zero, Cos x squared approaches 1.
But sin x squared approaches zero.
So the whole expression tends towards infinty.

If you're confused think of dividing 1 by smaller and smaller numbers till you divide it by such a small number that it almost reaches infinity (P.S. infinity is unreachable.)
Eg: 1/0.00000001, 1/0.0000000000000001 etc.

Again the expression is undefined at x= 0 since 1/0 is undefined.

Answer 6

oh yeah i forgot to mention the limit from the positive direction

Answer 7

so its more like 1/x as x approaches 0, Right Guys?

Answer 8

no, it's more like \[\frac{1}{x^2}\] as x approaches 0. this is because coming from the left or the right gives the limit.
\(\frac{1}{x}\) has different limits from the left and the right.

Answer 9

THANK YOU @pgpilot326 , @bloopman & @Navket_Jha

Answer 10

1/x -> -infinity from the left and +infinity from the right

1/x^2 -> infinity from the left and the right

Answer 11

you're welcome!

Answer 12

Anytime frnd. Close your question if done.