how cot is valid if 1 / infinity =/= 0

[Vastor]

in calculus, I learned that 1/infinity will approaches 0, but not equal to 0

 

like 1/0 is approaching infinity, but not equal to infinite

 

tan 90 = sin(90)/cos(90) = 1/0 = infinite?

 

1/tan 90 = 1/infinite = 0?

 

how does cot graph is valid (instead of using the "undefined" value or line up an asymptote there) just because the assumed value

of 1/tan(90) = 0, while tan 90 = infinite..

[John]

While tan(x) increases without bound as x approaches 90 degrees, tan(90) itself is undefined.

Therefore, thinking of cot(90) simply as 1/tan(90) isn't the way to go, since you're essentially trying to divide 1 by an undefined value. The better way to look at it is that since tan(x) = sin(x)/cos(x), then cot(x) = cos(x)/sin(x), which is defined at x = 90. Since cos(90) = 0 and sin(90) = 1, cot(90) = 0/1 = 0.

Edit: To elaborate a bit, when we say [math]\lim_{x\to{a}} f(x)=\infty[/math], we're simply saying that the value of f(x) increases without bound as x approaches a. In calculus, we actually say the limit doesn't exist in this case, but we can extend the real number line to include [math]\pm\infty[/math] and use these symbols as convenient shorthands, keeping in mind that they still aren't real numbers.

Edited February 5, 2014 by John