Problem 65 Hard Difficulty

Is there a number $ a $ such that
$$ \lim_{x \to -2}\frac{3x^2 + ax + a + 3}{x^2 + x - 2} $$
exists? If so, find the value of $ a $ and the value of the limit.

Answer

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Video Transcript

this problem Number sixty five of this store Calculus aesthetician section two point three Their number. Hey, such that eliminates X approaches. Negative, too. Of the function three X squared plus a experts able astri, divided by the quantity X squared plus X minus two exists. If so, find the value of a and the value filament. The reason there's concern whether this limit exists or not, is two to the denominator. Let's see what happened there only plaguing negative too. And two squared minus two minus two equals four minus two minutes to its hero. So then I made her a zero. And we have a concerned whether this exists or not run The only way that we're going to get ah, this limit equals zero or to exist is that even even though the denominator is zero, we'LL have an opportunity to have this limit exist. If the numerator is also zero. So when would that occur? I think this Oh no. And let's solve at negative too. And that sulfur value, eh, that makes this a function include a zero. Okay, so we have three times of forest talk, thanks to a plus. A plus three equals zero things to say is equal to NATO and moving that to the right side kiss positivity and we see that he is equal to fifteen. This is the number that we need in order for this limit to exist. So this is a part one year from the value of a a mystical fifteen because that makes the neuron are equal zero. Now, how do we find the value the limit? Well, we take the new polynomial three minutes care it is three X squared A is fifteen points or fifteen x, and again in his fifteen plus three is eighteen. I want a better by this opponent Meal X squared plus X minus two. We'LL be able to evaluate estimate by simplifying these polynomial. For example, the denominator reduces to expose, to add factors to expose, to multiply it by express one and the new marina I should tractor as such Explosives too. Three AKs Pleasant night, right? Because we get three x squared plus x X plus name That's fifteen nine times two is eighteen. At this point, we can see that in next was two councils and this was intended because this term is the part that made the denominator zero. And we chose the appropriate value of eight to make ah, to have their be affect. Er for the numerator. Cancel out when exes equal to negative too, to know we're left with three X plus nine. Tired of my experience. One and the value in and negative too. Getting into six plus thing, all running at three. And this casus value, I think it a one over our limit.