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)=1+sin x / cos x; limx→π/2^− f(x); lim x→4π/3 f(x)

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with continuity. This problem says for the function G of X equal to the quantity of 2 plus cosine of X in quantity divided by sin of X, identify the intervals of continuity and compute the limit, which is going to be the limit as X approaches 0 from the right of G of X. We give 4 possible choices as our answers. For choice A, 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 infinity. 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 equal to minus infinity. For choice C, we have the open interval of in pi to the quantity of n + 1 in quantity pi for any integer n, and the limit is infinity. And 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 equal to minus infinity. So the first thing we need to do is identify the intervals of continuity for our function GFX. And if we look at G of X, we notice that we have the sin of s on the denominator. So this function G of X will be continuous everywhere except when sin of X is equal to 0. To recall that sin of X, is equal to 0, When x is equal to some integer multiple of pi, which will write as n multiplied by pi where n is any integer. So that could be a positive or negative integer. So, that means that our function G of X is going to be continuous in between these integer multiples of pi. And so we can write that as the open interval. Going from in pie. 2, the quantity of N + 1 end quantity multiplied by pi. And this is actually going to be multiple intervals of continuity, since we'll have multiple values of N. So those are our intervals of continuity, and now we need to calculate our limit. And we're asked to calculate the limit. As X approaches 0 from the right of G of X, which is the quantity of 2 plus cosine X in quantity, divided by sin of X. So to evaluate this limit, we're going to look at the limit of our numerator and denominator separately. Now, if we evaluate the limit as X approach 0 from the right of 2 plus cosine X. Our 2+ cosine x is going to approach the value of 3 as X approaches 0 from the right. However, on the denominator here, we'll have X approaching 0 from the right of sin of X. So sin of x is going to be approaching the value of 0. So here we're going to be dividing by 0. We'll have 3 divided by 0. So this function is actually going to be approaching either plus or minus infinity, since our denominator is going to be getting smaller and smaller, so our overall value will be getting larger and larger. Now we're approaching x equal to 0 from the right and so. From the right, our sin of X is actually positive, so its value is actually greater than 0. So that means our denominator, as we approach um X equal to 0 from the right, is going to be a positive number. So we'll have a positive number divided by a positive number. So overall, we're going to get a positive value. So that means that this limit is going to be approaching. Positive infinity. And so that is the limit that we're looking for. And now that we have both our limit and intervals of continuity, we can look at our answer choices, and the correct answer choice corresponds to answer choice C. So I hope that this explanation has been useful, and I'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)=1+sin x / cos x; limx→π/2^− f(x); lim x→4π/3 f(x)

Key Concepts

Continuity of Functions

Limits

Trigonometric Functions

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