Find the limits in Exercises 31–40. Are the functions continuo...

Question

Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?

lim ϴ → 0 cos (πϴ/sin ϴ)

Accepted Answer

Welcome back, everyone. Find the limit as x approaches 0 of the expression sequence of pi x divided by sin of x. Determine if the function is continuous at the point being approached. So for this problem, let's begin with the limit. We want to identify the limit as x approaches 0 of sequent of pi X divided by sin of x. Now let's recall whats and S. We want to evaluate the limit as X approaches 0 of 1 divided by. Cosine of 5 x divided by sin of x. Now from here we're going to apply the properties of limits. We can distribute the limit, which gives us limit as x approaches 0 of 1 divided by limit as x approaches 0 of cosine of 5 x divided by sin of X, and furthermore, we can use another property which says that we can seek. The function of a limit, right? So in this case we have cosine of pi x divided by sin of x within the limit. We can take cosine of the limit value if the limit exists, right? So we're going to assume that and we're going to write limit as. X approaches 0 of 1 in the numerator for the denominator, we're going to take out cosine and evaluate cosine of the limit as x approaches 0 of x divided by sin x. Now let's go ahead and evaluate each limit. So for the numerator, limit SX approaches 0.1 is simply 1 because it is a constant which is independent of the X value, and now for the denominator, we're taking cosine of a limit. Now how do we evaluate that? Well, first of all, we can factor out pi, and we're getting cosine of pi multiplied by a limit. As X approaches 0 of X divided by sin x. And then we have to recall that this limit as X approaches 0 of x divided by sin of x or sin of x divided by X is equal to 1. So we are essentially evaluating 1 divided by cosine of pi multiplied by 1 or simply cosine of pi. Which gives us 1 divided by -1 and the answer is -1. So we have evaluated the limit. Let's label it as the answer. And now if we want to make sure that our function is continuous at that point, we simply want to check if we can evaluate the value of the function at X equals 0, right? So we're going to take 2. Of pi multiplied by 0 divided by sin of 0. Now, does it exist? Well, essentially, if we look at the sequin function. We are taking 0 divided by 0, right? Essentially division by 0 is undefined. Let's recall that sign of 0 is 0, and we cannot substitute 0 into the denominator because for a rational function we cannot divide by 0, right? And therefore, the value of the function. At X equals 0 is simply undefined, meaning the function is not continuous at X equals 0. So we have our limit and we can say that no. The function Is not. Continuous At X equals 0, because this leads to our denominator being equal to 0. That would be our final answer, and thank you for watching.

Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?
lim ϴ → 0 cos (πϴ/sin ϴ)

Key Concepts

Limits

Continuity

Trigonometric Limits

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