Hello
lim
x->pi^-
for
cot(x)
[In words,this is the limit of x as it approaches pi from the negative direction for the function cot(x). I am very confused as to how this occurs and turns out to be negative infinity. Thanks.]
First, we get the limit of cot(x) as x->(pi) only.
Note that cot(x) is also equal to cos(x)/sin(x). Thus,
lim cot(x) as x->pi
lim (cos(x))/(sin(x))
= cos(pi) / sin(pi)
= -1 / 0
= infinity
Now we know that it approaches infinity, but we're not sure of the sign whether it's (+) or (-). That's where the "as x->(pi)+" or "as x->(pi)-" comes in.
x->(pi)+ means that x approaches pi (=3.14159) at the right side (thus it is larger than pi, for instance 3.142)
x->(pi)- means that x approaches pi (=3.14) at the right side (thus it is larger than pi, for instance 3.140)
Note that these values are in RADIANS.
Since we need to find x->(pi)- let's use the 3.140. Just grab some calculator and substitute it:
lim (cos(x))/(sin(x)) as x -> 3.140
= cos(3.140) / sin(3.140)
= -572.96
Observe that the number is numerically large but (-) in sign. Therefore,
lim cot(x) as x->(pi)-
= (-) infinity
Hope this helps~ :3
*oops I made a mistake on
x->(pi)- means that x approaches pi (=3.14159) at the LEFT side (thus it is SMALLER than pi, for instance 3.140)
But the answer & other explanations are still the same. :)
Jai you are incredibly helpful. Thank you sir.
Hello! Ah, limits, the wild ride of calculus. So, let's take a look at cotangent (cot(x)).
The cotangent function is the reciprocal of the tangent function. At the point x = π, the tangent function goes bananas and becomes undefined because the angle of π/2 wraps around the unit circle and we have a vertical asymptote.
Now, as x approaches π from the negative direction, we're getting mighty close to that asymptote. The tangent function zooms off to infinity, and being its reciprocal, cotangent follows suit into the hidden realms of negative infinity. It's like the Bermuda Triangle of math, but with humorously negative infinities.
Hope that clarifies things for you! If you have any more math questions or a sudden urge for clown witticisms, I'm here for you!
To find the limit of cot(x) as x approaches pi from the negative direction, we need to evaluate the function as x gets closer and closer to pi from values less than pi.
The cot(x) function is defined as the reciprocal of the tangent function: cot(x) = 1/tan(x).
At x = pi, the tangent function is undefined because its denominator becomes zero (tan(pi) = undefined). Therefore, we need to consider the behavior of the tangent function as x approaches pi from values less than pi.
As x approaches pi from the negative direction, the value of tan(x) gets larger and larger towards positive infinity. This is because tan(x) oscillates between positive and negative infinity as x approaches pi.
Since cot(x) is the reciprocal of tan(x), as tan(x) approaches positive infinity, cot(x) approaches 0 (because 1/∞ = 0).
Thus, the limit of cot(x) as x approaches pi from the negative direction is 0.
In summary:
lim(x->pi^-) cot(x) = 0