Continuous Extension

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Question

Continuous Extension

Explain why the function ƒ(𝓍) = sin(1/𝓍) has no continuous extension to 𝓍 = 0.

Accepted Answer

Below there, today we're going to 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. Determine whether GFX equals cosine of 2 divided by X as a continuous extension at X equals 0, justify your answer. So it appears for this particular problem, we're asked to determine whether or not GFX equals cosine of 2 divided by X, we'll have a continuous extension at X equals 0, and we're asked to justify our answer. So now that we know what we're solving for, let's take a moment here to read off our multiple choice answers to see what our final answer might be. A is GFX equals cosine of 2 divided by X has a continuous extension at X equals 0, because the limit as X approaches 0 of, cosine of 2 divided by X does not exist. B is G of X equals cosine of 2 divided by X has a continuous extension at X equals 0, because the limit as X approaches 0 of, cosine of 2 divided by X is equal to 0. C is G of X equals cosine of 2 divided by X has no continuous extension at X equals 0, because the limit as X approaches 0 of cosine of 2 divided by X is equal to 0, and finally, D is G of X equals the cosine of 2 divided by X has no continuous extension at X equals 0, because the limit as X approaches 0 of cosine of 2 divided by X does not exist. Fantastic. So, first off, in order to solve this problem, we need to recall and note that a continuous extension of a function. At a point means that we can define the function at that point in a way that makes it continuous. Specifically, we can say that if a function G of X is not originally defined at X equals A, we can attempt to define GA such that the limit as X approaches A. Of G of X is equal to G A. And once again, this means, once again, we're trying to say that specifically a function G of X is not defined at X equals A. So we can attempt to define G A such that once again the limit as X approaches A of G of X is equal to G A. So, with that said, If this is possible, then we can say that the function has a continuous extension at X equals A. So, with that said, in order to check whether a function has a continuous extension at a point, we need to determine if the limit of the function exists at that specific point. If the limit exists, we can define the function at the point to match the limit, which will therefore make the function continuous. However, it's important to note that if the limit does not exist, a continuous extension will not be possible. So with that in mind, we want to determine whether this function of G of X is equal to cosine of 2 divided by X has a continuous extension at. X equals 0. So we're trying to figure out if it has a continuous extension at X equals 0. So now, with that said, we need to check the limit at X equals 0. So for G of X to have a continuous extension at X equals 0, the limits, we need to say that the limit as X approaches 0 of, cosine of 2 divided by X. Must exist. So with that said, We can therefore state and note, however, that as X approaches 0, we need to note that the expression 2 divided by X will grow arbitrarily large in magnitude. And the cosine function will oscillate between -1 and 1 as the input changes. So therefore, since 2 divided by X keeps increasing and decreasing without bound, that will mean that this cosine of 2 divided by X will oscillate indefinitely between -1 and 1, and because of these rapid oscillations, that will mean as a result, G of X does not settle at a single value as X approaches 0, meaning that the limit as X approaches 0 of the cosine of 2 divided by X does not exist. So DNE does not exist. So it does not exist. So it's very important, make an explanation point, does not exist. So, therefore, without a doubt, since this limit does not exist, that means that G of X is equal to the cosine of 2 divided by X has no continuous extension, which I'll denote as CE, so it has no continuous extension at X equals 0, so no continuous extension. So therefore, without a doubt, our final answer has to be multiple choice answer letter D, which once again states that G of X is equal to cosine of 2 divided by X, has no continuous extension at X equals 0, because the limit as X approaches 0 of cosine of 2 divided by X does not exist. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Continuous Extension

Explain why the function ƒ(𝓍) = sin(1/𝓍) has no continuous extension to 𝓍 = 0.

Key Concepts

Limits

Continuity

Oscillation

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