[Technology Exercise] In Exercises 33–36, gra...

Question

[Technology Exercise] In Exercises 33–36, graph the function to see whether it appears to have a continuous extension to the given point a. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at a. If the function does not appear to have a continuous extension, can it be extended to be continuous from the right or left? If so, what do you think the extended function’s value should be?

g(θ) = 5 cos θ / (4θ ― 2π) , a = π/2

g(θ) = 5 cos θ / (4θ ― 2π) , a = π/2

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Consider the function H of X is equal to. 3 multiplied by sin of 2 X, all divided by 2 X minus pi, for X cannot equal pi divided by 2. Using Lajapatal's rule, determine the value of H X as X approaches pi divided by 2 for a possible continuous extension at X equals pi divided by 2. Awesome. So it appears for this particular problem, we're asked to use Lajapatal's rule to help us determine the value of this function of H X as X approaches pi divided by 2 for a possible continuous extension at X equals pi divided by 2. So now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be. A is 3, B is 0, C is -3, and D is the limit does not exist. Awesome. So first off, let us recall and note that a continuous extension of a function at a point means that we can define the function at that specific point in such a way that it makes it continuous. Specifically, as we should note and recall, if a function H of X is not originally defined at X equals A, we can attempt to define H of A such that the limit as X approaches A of H of X is equal to H of A. Fantastic. So with that in mind, we can say that if this is possible, then we can say that the function has a continuous extension at X equals A. So therefore, the limit that we are trying to evaluate will be the limit as X approaches pi divided by 2 of H of X is going to be equal to the limit as X approaches pi divided by 2 of 3 multiplied by sin of 2 X divided by 2 X minus pi. Fantastic. So now, our next order of business is we need to recall and recognize the fact that If we use direct substitution of X equals pi divided by 2, this will lead to an indeterminate form as an answer, which, as we should recall, let's make a note here, up above, up above here, let's make a note. So an indeterminate form means when you do the direct substitution method, meaning since we know that X approaches pi divided by 2, if we plug that value of pi divide by 2 in for all of our values for X and we evaluate and solve, that will mean that we get an indeterminate form, which means we get an answer of 0 divided by 0 or infinity divided by infinity, and because we get an indeterminate form when we use a direct substitution method, that means we can use Lajapatal's rule. So because we know that the direct substitution of X equals pi divided by 2 leads to an indeterminate form as an answer, this means that we will be using Lajapatal's rule to evaluate the specific limit. So that means when we plug in, so let's plug in a value of Pi divide by 2, N for X. So if we take 3 multiplied by sign of 2 multiplied by pi divided by 2. Divided by drum roll, 2 multiplied by pi divided by 2 multiplied by, or actually it's gonna be 2 multiplied by pi divided by 2 minus pi. So when we evaluate that expression, we will get 3 multiplied by 0, divided by 0, which gives us 0 divided by 0. As an answer, which once again, since we have an indeterminate form as an answer, we have to apply Lajapatal's rule in order to solve for this particular limit. So in order to apply Lajapa's rule, as we should recall, we need to differentiate the numerator and denominator separately with respect to X, and after applying Lahapatal's rule, we need To note that if the expression still results in an indeterminate form, we will need to repeat this differentiation process until a determinant form is achieved, meaning as long as we don't get 0 divided by 0 or infinity divided by infinity as an answer, that means that we found a determinant form which that answer that doesn't result in an indeterminate form as an answer will be our final answer for the evaluated limit. So with that said, we need to recall that D by DX of sine of X will equal cosine of X. And we also need to recall. That we can go ahead and write that the limit as X approaches pi divided by 2 of D by DX of 3 multiplied by sin of 2. X divided by D by DX of 2 X minus pi. is going to be equal to drum roll, the limit as X approaches pi divided by 2. So if we say the limit as X approaches pi divided by 2 of, and when we perform the derivative, we'll get 3 multiplied by cosine of 2 X multiplied by 2. So 3 multiplied by cosine of 2 X multiplied by 2, all divided by 2, which when we simplify, we'll get that the limit as X approaches pi divided by 2. Of 3 multiplied by cosine of 2 X. is going to be equal to 3, multiplied by the limit as X approaches pi divided by 2 of the cosine of 2 X. Fantastic. We are on a roll. So, our next step now, since we know that when we have it simplified, that we have 3 multiplied by the limit as X approaches pi divided by 2 of cosine of 2 X. We now need to use our direct substitution method, so we need to evaluate the limit as X approaches pi divided by 2. So, with that said, Let us note that when we do the direct substitution method, meaning if we take 3 multiplied by the limit, as X approaches pi divided by 2 of cosine of 2 X, we will find when we do direct substitution, we will get square bracket 3 multiplied by cosine of 2 multiplied by pi divided by 2. Which will be equal to 3 multiplied by cosine of pi. Which means that it will be equal to drum roll. B square brackets 3 minus 3 multiplied by -1 or 3 minus 1. So 3 multiplied by negative. One, which will give us an answer of, so essentially you need to note that when we evaluate cosine api, that will mean, so it's not -1, it's specifically negative one because if you plug in, make sure your calculators in radiance form because if it's not in radiance, I should say radiance mode, if your calculator's not in radiance mode, that means you'll not get the correct answer. So when you put code Sign up pi into your calculator making shirts and ratings form. You'll get an answer of -1, so 3 multiplied by -1 will give us an answer of -3. So therefore, without a doubt, that since the limit exists and is finite, we can define H of pi divided by 2 is equal to -3. So let's make a note of that. So we can say that H of Pi divided by 2 is equal to -3 to make H of X continuous at X equals pi divided by 2. So that means without a doubt, our final answer has to be -3, and this corresponds with multiple choice answer letter C, which once again is -3. Thank you very much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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