What Value of a Will Make the Function Continuous Sinx 2x
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What value for a will make sinx ,X # 0 f(x) = 2x 'X =0 a continuous function?
Related Question
The function f is given by the formula 2x x2 _ x f(x) when x < 0 and by the formula f(x) = cos(x) + a when x > 0_ What value must be chosen for a in order to make this function continuous at x = 0 ? 2
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Video Transcript
in this question, we are given a piecewise defined function which depends on the parameter A. And were asked to choose to find the value of the parameter A. Which makes this function continues. So recall that the function is continuous is the limit of F of X. As X goes to a ah sorry as in our case it's going to be as X goes to zero is equal to f of zero. In this case our function is continuous at zero. So basically we want to find the value of A. Which makes this limit is equal to the value to F. Of zero. But first let's find F of zero. So to find F of zero, we need to follow the definition of the function and it says that for X greater equal than zero, we need to use the formula cosine X plus A and zero is included in this interval, right? Because it is greater equal than zero. Therefore we have to use the second row to find the value of F of zero. We're going to get costs zero plus eight. And since because zero is equal to one, we're going to get eight plus one. So fo zero is equal to eight plus one. Now we need to find the limit of F of X. As X goes to zero, limit as X goes to zero of F of X. And here we have to specify that we are going to zero from the left right. If you're going to zero from the left, this means that X is slightly less than zero and we are going to use the first row. So this limit is equal to the limit As X goes to zero from the left of two, X over x squared minus x. Now this limit has a form 0/0. However, we can we can change this uncertainty by factoring out X from the denominator in the denominator we're going to get X times x minus one As X goes to zero from the left. Now we can cancel accent the numerator and denominator. So we're going to get Limit of two over X -1 As X goes to zero from the left. Now plug in zero, We're going to get two over negative one And this is equal to -2. Now we need to find the limit of F of X when X goes to zero from the right hand side. Now when X goes to zero from the right hand side this means that X is slightly bigger than zero and we're going to use the second formula. So we need to find the limit of course X plus a. It's X. goes to zero from the right. But this is easy. You just need to plug in Mexico zero here. Right. So we don't have any like undetermined forms here. So we're going to get a cost zero plus eight which is gonna be one plus A. So for the for the function to be continuous, we want the two sided limit to exist and be equal to the actual value of the function. Well in our cases this is simpler because the limit from the right is equal to the actual value of the function. So we just need to solve the equation. One plus eight Is equal to the limit from the left, which is equal to -2. Now moving on to the right hand side, we're going to get A is equal to negative three, and this is the value of A. Which is going to make this function continuous.
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