evaluate-each-limit-if-it-exists-use-l-hospitals-rule-if-appropriate-lim-_x-rightarrow-2-fracx3-8x5-

Question

Evaluate each limit (if it exists). Use $L$ Hospital's rule (if appropriate).
$$\lim _{x \rightarrow 2} \frac{x^{3}-8}{x^{5}-32}$$

Carson Merrill · Accepted Answer

So there is a way to solve this without using little tiles role. We can actually factor X cubed plus eight um to be X plus two yes and x squared minus two acts class four or 2 squared. I mean as a result of that it allows us to cancel the X plus twos just leaving us with um This portion right here. That way when we evaluated at -2 we would end up getting 12 is our answer. Now in this case because it&#x27;s a cubic function, it would actually be easier to use low Patel&#x27;s rule and then we explain why all we have to do is take the derivative of the numerator and the derivative of the denominator. That would give us one on the denominator and it would give us three X squared on the numerator. Yeah squared. So with that alone we would end up being able to evaluate it at negative two and once again that would give us 12 as the final answer.

Amy Jiang · Answer

Okay, so we want to find her falling women about you. So it&#x27;s done its inner numerator. We have X cubed by his feet and eight skin beards. And as to two per three this year is the difference of two cubes. So we can practice into X minus one times X squared plus two acts plus four. And in our denominator, we have X to about four minus 16 which is, um, we can write this us forward to depart to, and I was, in fact, the routes A to that is we have extra part to depart to minus 4 to 5 to So this is a difference of two squares so we can practice into X minus four. Actually, that&#x27;s X squared. What is four times X squared plus four. And here we have the difference of two squares once again. So we get X might still Times X plus two and then we have X squared plus four. Let&#x27;s notice that we haven&#x27;t experts to in common that we can counsel out to see that we&#x27;re left with the limits as X approaches to of X squared plus two x, what&#x27;s for all over X plus two times X squared plus four and islets used Rex up. So we have two squared, which is four plus two times two, which is four plus warm all over two plus four, that&#x27;s for and then to square. That&#x27;s four plus four, which is eight. Okay, so we have four plus work was for its 12 over four times eight. Let&#x27;s recalled up 12 is three times for we can counsel out or four and we see that rule us with a 3/8.

Ernest Castorena · Answer

Okay, This time we have the function X to the fifth, minus 32 over X cubed minus eight. So for this rational function, they want us to know the limit as we approach to. Well, it&#x27;s clearly undefined that to cause that denominator will be dividing by zero at that point. So let&#x27;s approach to from the raid. So first will be 0.1 units to it on the rate then point 00 on distance to it on the name 00.1 and 0.1 00001 Well, the point is, we&#x27;re getting closer and closer to two, and we can see Oh, it looks like we&#x27;re pro Gene six and 2/3 and everybody from the left. So a negative 0.1 unit from two. So I left and then we&#x27;ll get closer and closer to it from the left, and it looks like we are in need approaching six and 2/3 so we can say that our limit appears to be, Let&#x27;s see, what&#x27;s another eight or 86 2/3. Probably like funny over there. So something over three years, aren&#x27;t they awesome?

Amy Jiang · Answer

so we want to find their following limits. Let&#x27;s look better falling function here. We&#x27;ll notice that this year is a polynomial fillets use or limits Roll number six so we can plug in our ex points into are falling function. So we&#x27;re gonna have five times. One could part eight minus three times one do you Part two plus two. So that&#x27;s just people to five minus dream. What&#x27;s to do? We get to go still, which is equal to four.

Carson Merrill · Answer

So you see this problem here has two options, one with and one without little tells role. First option is if we factor the numerator, we see that. This is the same thing as X plus two Times Active Plus four. And that allows us to cancel with something in the denominator because right down here we have exposed to in the next -4. So based on this we can cancel the X -4. Um and the X minus for giving us X plus tours are line. So when we plug in for um for the limit, we just end up getting six as our answer. But now let&#x27;s try doing this with the help of cows rule. So basically here, if we plugged in X equals four, we would get For for this would give us 0/0, which is an indeterminate form. So now we have to take the uh We have to use locales role giving us one and the nominator and up in the numerator we would end up getting two X -2. So when we plug in four here, that would be 8 -2, which is 6/1, that&#x27;s six. So once again we get a final answer of six.

Carson Merrill · Answer

Okay, The question is fine. The following live it use locked US rule. Oh, maybe Amara elementary methods. So first, let&#x27;s see this limit. This is a fraction and maybe the first of thing tto find the limit iss plucking so we can plug in minus two in tow. This formula. So that isthe. First, let&#x27;s look at the top that helped ISS minus two cube that isthe minus eight. Plus it so is zero, and the bottom is minus two plus two is also zero. So we know that zero divided by zero. We can use the laptops rule to say that this limit isi coto the limit as X goes to minus two the purity of off the top which is three x square and curative off the bottom is what So that is Seiko to you. Plucky minus two in tow. This easier for Miller, that is three times minus two square is for so is twelve. Maybe you can also think this one more elementary methods. Ah, look at that Hop it Issa polynomial function and about him is also the polynomial and it hot. The degree off the top is three on the bottom is one so we can use the long division tto find another factor off the top if one the fact wealth the factors is X Class two. So if we do that, we know that X Square plus eight. We know what affect her ISS X Plus two. And after the calculation, you will know that the other factor is X Square minus two x plus fall. So if he writes the top as this the product off these two foot factors, we can cancel the same one X plus two. So this limit is also equal to the limit as X goes to minus two the top We laughed the X Square minus two x plus fall and the ball home we can So the X plus two. So that is what? And in this way you plug in minus two into this one so that this man is too square for you, minus two times minus two plus fall and plus four. So it is also twelve

Problem

Evaluate each limit (if it exists). Use $L$ Hospi…

Problem 33 Easy Difficulty

Evaluate each limit (if it exists). Use $L$ Hospital's rule (if appropriate).
$$\lim _{x \rightarrow 2} \frac{x^{3}-8}{x^{5}-32}$$

Answer

$\frac{3}{20}$

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus