Determine the interval(s) on which the following functions are...

Question

Determine the interval(s) on which the following functions are continuous; then analyze the given limits.

f(x)=csc x;lim x→π/4f (x);lim x→2π^− f(x)

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with continuity. This problem says for the function H of X is equal to coten of X, identify the intervals of continuity and compute the limit as X approaches pi divided by 6 of H of X. We're given 4 possible choices as our answers. For choice A, we have the open interval of n pi 2, the quantity of n + 1 in quantity multiplied by pi for any integer n, and the limit is the square root of 3. For choice B, we have the open interval of n pi divided by 22, the quantity of n + 1 in quantity pi divided by 2 for any integer n, and the limit is the square root of 3. For choice C we have the open interval of n pi divided by 22, the quantity of n + 1 in quantity, pi divided by 2 for any integer n, and the limit is 1 divided by the square root of 3. And finally for choice d we have the open interval of n pi to the quantity of n + 1 in quantity pi for any integer n, and the limit is 1 divided by the square root of 3. So, the first thing we need to do in this problem is to identify the intervals of continuity. And so we're looking at the function of cotangent of X. And so recall that the cotangent of X is equal to the cosine of X. Divided by the sin of X. So that means cotangent of X is continuous anywhere sin of x is not equal to zero. And so recall that the sin of X. Is equal to 0. When X is equal to some integer multiple of pi, so we'll write that as N multiplied by pi, wherein here is an integer. And so that it's going to be for any integer. So that means our function is continuous in between these um integer multiples of pi. So we can actually write that as the open interval. Of in pie. 2, the quantity of N + 1 multiplied by pi. So those are gonna be our intervals of continuity. So we have multiple in values, we're gonna have multiple intervals of continuity. Now, the second part of this question asks us to calculate the limit as X approaches pi divided by 6. Of our function H of X, which is cotangent of X. Now, since the um value of pi divided by 6 falls within one of our intervals of continuity, then we can simply um to evaluate this limit, since our function is continuous, we need to simply evaluate our function at the value of X that we're trying to find the limit for. So that means that our limit here, as X approaches pi divided by 6 of cotangent of X, is going to be equal to the cotangent. Of pi divided by 6. And we can evaluate the express expression, and this is going to be equal to the square root of 3. So that is the other half of the answer that we're looking for. And so now, if we look at our answer choices, the correct answer choice actually corresponds to answer choice A. So I hope that this explanation has been useful, and we'll see you in the next video.

Determine the interval(s) on which the following functions are continuous; then analyze the given limits.

f(x)=csc x;lim x→π/4f (x);lim x→2π^− f(x)

Key Concepts

Continuity of Functions

Cosecant Function

Limits

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