Find functions f and g such that lim x→1 f(x)=0 and lim x→1 (f...

Question

Find functions f and g such that lim x→1 f(x)=0 and lim x→1 (f(x)g(x))=5.

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, find functions H and K such that the limit as Y approaches two of H of Y equals zero and limit as Y approaches two of H of Y multiplied by K of Y is equal to eight. Awesome. So it appears for this particular problem we're trying to find the functions for H and K given these two different limits. So now that we know what we're ultimately trying to solve for, let's take a moment here to read off our multiple choice answers to see what our final answer pair might be noting that we're given an answer for the function of H of Y first. And then we're given an answer for the function for K of Y. So and they're gonna be H of Y is equal to sum expression and K of Y is equal to some expression. So A is Y minus two and eight divided by YB is Y plus two and eight divided by Y plus two C is parenthesis, Y minus two parenthesis and eight divided by Y minus two. And finally, D is Y and eight divided by Y plus two. OK. So first off in order for us to effectively and efficiently solve this particular problem or any problem like this one, we need to define our known variables and our target variable. So we need to find the functions of H of Y and K of Y. So let's write that down. So we're trying to figure out H of Y and we also need to figure out so H of Y and K of Y, so we need to figure out these two functions. So K of Y and H of Y Fantastic. And we're also given, we're given these two different limits. We're given the limit as Y approaches two of H of Y. And we're also given, oh And don't forget that the limit as Y approaches two of H of Y is equal to zero. And we're also told the limit where the limit of Y approaches to, of H, of Y multiplied by K of Y is all equal to eight Fantastic. So with that in mind, we now need to move on to our next step. We need to select that H of Y, we need to choose an H of Y such that the limit as Y approaches two of H of Y equals zero. So let us say that, that H of Y. So let's go ahead and say that H of Y and we're gonna say that this H of Y is equal to Y minus two. And we also need to determine a value of K of Y such that the limit as Y approaches two of H of Y multiplied by ky equals eight. So to do that, we need to substitute our value, which is H of Y is equal to Y minus two into our second limit. And when we do that, we will be able to write that the limit as Y approaches two of parenthesis, parenthesis, Y minus two. And we need to multiply this by K of Y and then this will be all equal to eight and we need to solve for K of Y. So when we do that, we will find that K of Y is equal to eight divided by Y minus two. So that's when we solve for K of Y using our limit expression. So now our next step is we need to verify our limits. So let us note that for H of Y, so H of Y, this will be equal to Y minus two. And then we need to do comma so like call, you know, call and comma. So this is gonna be once again. So H of Y equals Y minus two comma and then we need to focus on our limit, which is Y. So our limit as Y approaches two of H of Y and then don't forget that H of Y. So this is gonna be H of Y. So the limit, once again, the limit, as Y approaches two of H of Y is gonna be equal to our limit of or our limit where Y once again approaches to, of Y minus two, so Y minus two and this will be all equal to zero. And similarly, so for K of Y, so K of Y is equal to eight divided by Y minus two, our limit as Y approaches two of H of Y multiplied by K of Y is all equal to. So it's gonna be all equal to the limit of Y as Y approaches to of parenthesis, parenthesis, Y minus two multiplied by eight divided by Y minus two. And this will be all equal to eight. Fantastic. So taking a breath here. So therefore, without a doubt, the functions are, and let's write our final answers in green. So for H of Y, our function will be equal to. So H of Y is equal to Y minus two and for K of YK of Y is going to be equal to eight divided by Y minus two. And these are our final answers. Hooray, we did it. So looking at our multiple choice answers, our final answer pair has to be multiple choice answer letter C which once again is H of Y equals Y minus two and K of Y equals eight divided by Y minus two. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Find functions f and g such that lim x→1 f(x)=0 and lim x→1 (f(x)g(x))=5.

Key Concepts

Limits

Product of Limits

Finding Functions

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