Evaluate each limit and justify your answer.
lim x...

Question

Evaluate each limit and justify your answer.

lim x→5 ln 6(√x^2−16−3) / 5x−25

Accepted Answer

Welcome back, everyone. In this problem, we want to evaluate the limit of the function F X equals the natural log of 12 multiplied by the square root of X2 minus 64 minus 6, all divided by 6 X minus 60 as X approaches 10. A says it's the natural log of 10, B the natural log of 3/10. C the natural log of 10/3, and D says the limit does not exist. Now for our function, if we go ahead and substitute the value of X equals 10 into it to solve for the limit, notice that it will make the denominator 0. So we'll have to try to rewrite our function so that our function doesn't have a denominator of 0, making it undefined to help us figure out the limit. Now, let's try to do that by first rationalizing the numerator, OK? So rationalizing the numerator. Of FFX, OK. Which means that we are going to multiply it by the conjugate of the term in the numerator, OK. Then we can write FFX as the natural log. Of 12 multiplied by the square root of X2 minus 64 minus 6 all divided by 6 X minus 60 multiplied by the square root of X2 minus 64 plus 6. Divided by the square root of X quad minus 64 plus 6. Now when we go ahead and do that. That is going to be the natural log of 12 multiplied by these terms, OK, the skirt of X minus 64 minus 6 multiplied by the skirt of X minus 64 + 6. And no, since there's the difference of two squares, we're going to be multiplying here. The square of X2 minus 6 square root of X2 minus 64, which would just be X2 minus 64 minus the square of 6. So that's minus 36. Again, all of that is being divided by 6 X minus 60 multiplied by the conjugate. No, we can simplify this. We can take this a step further, OK, because. 6 X2 minus 64 minus 36 could be written as X2 minus 100. OK. And now notice again in our numerator, we have a term that has the difference of two squares. So in this case, we could simplify X2 minus 100, then we could rewrite it, OK. As x minus 10 multiplied by X + 10. Likewise, in our denominator, we can factor 6 from 6 X minus 60 to have it as 6 multiplied by X minus 10, multiplied by our conjugate term, OK? And again, I, I forgot to put the brackets here, so the parenthesis here, sorry, so please forgive me, OK, but that's how we can write it. OK. And no, notice that we can cancel X minus 10 from the numerator and the denominator and we know 6 goes into 12 2 times. So we can then write F of X as the natural log of 2 multiplied by X + 10. Divided by the square root of X2 minus 64. Plus 6 Now we can go ahead and solve for or limit by substituting x into the simplified function. Because no, this means then. That the limit, OK. As X approaches 10 of F of X. Is going to be equal to the limit as X approaches 10. Of oh, sorry, the natural log, OK. Off our expression And if we substitute, OK, if we substitute 10 into our expression, then this is going to be equal to the natural log of 2 multiplied by 10 + 10. Divided by the square root of 102 minus 64, OK, plus 6. No, let's simplify. In our numerator, we know that 10 + 10 is 20 and 2 multiplied by 20 equals 40. In the denominator, 10 squad is 100, 100 minus 64 equals 36, and the square root of 36 is 6, where 6 + 6 equals 12. So it simplifies to the natural log of 40 divided by 12, and since both of them have 4 as a common factor, 4 goes into 4010 times and into 12 3 times. Thus, the limit of F of X as X approaches 10 is going to be the natural log of 10/3. If we look back at our answer choices, that tells us C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Evaluate each limit and justify your answer.
lim x→5 ln 6(√x^2−16−3) / 5x−25

Key Concepts

Limits

Natural Logarithm

Indeterminate Forms

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