Evaluate each limit.

lim x→e^2 ln^2x−5 ...

Question

Evaluate each limit.

lim x→e^2 ln^2x−5 ln x+6 lnx−2

Accepted Answer

Welcome back, everyone. Evaluate the limit as x approaches E to the power of 3 of the function F of X is equal to ln2 x minus 4 LNX plus 3 divided by LNX minus 3. We're given 4 answer choices A says 0, B4, C2, and D does not exist. So let's define the limits. We have limit as X approaches E cubed. Of F of X, so that's LN squared. Of X minus 4 LN of X plus 3. Divided by a n of x minus 3, we're going to attempt the direct substitution, and if we substitute X equals e cubed, we're going to get 0 in the numerator and 0 in the denominator. So what we want to do is simply eliminate this in the terminal form, and we can do that by using factorization. So first of all, Let's assume that a lot of X is Z. Then our numerator becomes z quad minus 4 z + 3, and this is a quadratic polynomial we can factor it out as Z minus 3 multiplied by Z minus 1 substituting the natural logarithm back. We can say that our numerator is simply the product between. LN of x minus 3 and LN of x minus 1 and in the denominator we have that common factor LNX minus 3, so we can cancel out the whole of this rational function. And now we can attempt the direct substitution once again. We only have LN of X minus 1 remaining, so we're going to take LN. Of E to the power of 3rd, we can use the power rule, that's simply 3, right, minus 1. And 3 minus 1 gives us 2. This is now a finite number, so we can say that the answer to C corresponds to the correct answer. Thank you for watching.

Evaluate each limit. lim x→e^2 ln^2x−5 ln x+6 lnx−2

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Evaluate each limit.

lim x→e^2 ln^2x−5 ln x+6 lnx−2