Find the function The following limits repres...

Question

Find the function The following limits represent the slope of a curve y = f(x) at the point (a,f(a)). Determine a possible function f and number a; then calculate the limit.

(lim x🠂1) 3x²+4x-7 / x-1

Accepted Answer

Welcome back, everyone. In this problem, we want to find a possible function H of X and the number C such that the following limit represents the slope of the curve Y equals HX at C H C. Then we want to evaluate the limit, that is the limit as X approaches -1 of 4 X2 minus 2X minus 6 all divided by X + 1. Now what do we know here that's going to help us find H of X, C, and our limit? Well, recall, OK, that if we are given Y to be equal to H of X, OK. At that point C H of C, OK. Then the derivative of H of X at that point, that is H C, OK, can be obtained by using the equation that tells us it will be equal to the limit as X approaches C of H X minus H of C, all divided by X minus C. Now if we compare both our limits, what this is telling us is that if by comparison, we can find a way to get H of X and C, then we should be able to go ahead and solve with so sorry to, to figure out the limit. Now, let's go ahead and do that. So, if we compare, OK, if we compare both of our expressions, OK. Then we are saying. That H X minus H of C, OK, divided by X minus C is going to be equal to 4 X 2 minus 2X minus 6 divided by X + 1. I know if that's the case, OK, this is gonna be equal to X + 1. Then, again, if we do a bit of rewriting here, we could write this as 4 X squared minus 2 X, OK? minus 6 all divided by X minus -1. Now, when you compare, what do you notice? We'll notice here that 4 X2 minus 2X is a function of X, H of X, and in our denominator X minus C, notice here that C is -1. So from this, OK, we can infer. That 8 of X is equal to 4 X squad minus 2 X. OK. C equals -1. And thus H of C, which would be H of -1. Should be equal to 6. And now when we think about it, that makes sense because 4 multiplied by -1 squad is 4, minus 2 multiplied by -1, that's plus 2, and 4 + 2 equals 6. OK. So now we're sure that H of X is 4 X2 minus 2 X. And that C equals -1. Now we know those values, we can find a derivative. Of H at that point, that is when C is -1. Cause no, that means, again, based on what we know, it's gonna be the limit as X approaches -1 of 4 X square minus 2X minus 6 divided by X + 1. Now, notice here, if we directly substitute -1 into this limit in the denominator, it's gonna be -1 + 1 which equals 0. So we have to find a way to rewrite, or let's see if we can find a way to rewrite our numerator to get a common factor that we can cancel. Not for 4 X squared minus 2X minus 6. We could rewrite this, OK. I mean, as X approach is -1, as 2 multiplied by X + 1 multiplied by 2 X, OK, minus 3. And know if we have X +1 in our denominator and numerator we can cancel to be left with our limit. As X approaches -1 of 2 multiplied by 2 X minus 3. Now when we directly substitute, that's 2 multiplied by 2 multiplied by -1 minus 32 multiplied by -1 is -2, and -2 minus 3 is -5, 2 multiplied by -5 equals -10. Thus our limit as X approaches -1 or 4 X2 minus 2X minus 6 all divided by X + 1 is negative 10. H X is 4 X2 minus 2X and C equals -1. Thanks for watching everyone. I hope this video helped.

Find the function The following limits represent the slope of a curve y = f(x) at the point (a,f(a)). Determine a possible function f and number a; then calculate the limit.
(lim x🠂1) 3x²+4x-7 / x-1

Key Concepts

Limits

Derivatives

Function Evaluation

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