Limits and Continuity

On wha...

Question

Limits and Continuity

On what intervals are the following functions continuous?

b. g(x) = csc x

Accepted Answer

Welcome back, everyone. On what interval is the following function continuous FFX equals cotangent of X. So let's recall that cotangent is simply the ratio between cosine and sign. So we can treat this function as a rational function. And for a rational function, it is undefined whenever our denominator is equal to 0. So we want to make sure that sin of x is not equal to 0, which will represent. The continuity of the function, right, since we're basically finding the domain and we have to recall that sin of x is equal to 0 when X is a multiple of pis such as pi multiplied by n where n belongs to integers. So we are going to say that the domain of this function is X not equal to pi N where N belongs to integers, but we want to write it in an interval notation, and what we have to do is simply recall that cotangent is a periodic function, right? So we're going to analyze. Several periods. Let's suppose that we take negative pi and 0, so it's not defined as negative pi, it's not defined as 0, followed by i, 2 pi, and so on, right? So for every interval between 0 and pi, between negative pi and 0, between pi and 2 pi, our function is defined, right? So we're going to say that our function is continuous if X belongs to the interval from. N multiplied by pi up to N + 1 pi. It should be an open interval because our cotangent function is undefined at the extremes, right? And we are basically saying that for any integer M. Or basically for any um That belongs to integers. Our function is continuous for a multiple of pi up to that multiple plus 1 multiplied by pi. We can see if that's correct. Let's suppose that we're taking an interval from 0 to pi. In this case, our N is 0, right? So if N is 0, we get from 0 up to pi because 0 + 1 gives us 1. If our. N is equal to 1, then we get from pi up to 2 pi, which is exactly what we can see. So this interval notation basically excludes all of the pi multiples from the domain, and that's it. We have our final answer. Thank you for watching.

Limits and Continuity

On what intervals are the following functions continuous?

b. g(x) = csc x

Key Concepts

Continuity of Functions

Cosecant Function

Intervals of Continuity

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