Find the limit, if it exists and prove that number is the limit; otherwise_ prove that the limit does not exist.lim(")=o) 272 + 342 24 lim (J)-() 212 342...

Find the limit, if it exists, or show that the limit does not exist.
$\lim _{(x, y) \rightarrow(2,1)} \frac{4-x y}{x^{2}+3 y^{2}}$

All right, we want to find a limit of this expression as X approaches positive infinity. Since X is going to be approaching positive infinity, it's going to be positive and it's going to be very large and then larger and even larger. So when we look at the numerator, X to the fourth minus three X squared plus x. Even though we have three terms uh as X gets very very large uh one of these terms, the one with the highest power is going to be the dominant term, so the numerator is going to be dominated by extra fourth uh because as X is very large, exit 1/4 will be extremely large three times X to the second won't be anywhere near uh you know, as large as X to the fourth. So even though we're subtracting three X squared uh this whole numerator really is going to be dominated by the highest power backs D x to the fourth term. Likewise, the denominator is going to be not is going to be dominated by the X to the third term. So this entire expression is basically going to be dominated by X to the 4th over X to the third as X approaches positive infinity Now exit the 4th divided by extra 3rd is really x. Okay, so as X approaches positive infinity. This entire expression is basically going to behave as the value of X itself would. So it's X gets a very very large X or this expression this limit is also going to get very very large. So as X approaches infinity, we expect this function to approach infinity. Here on the graph. I have the function graft. And you can clearly see that as as X approaches positive infinity, the function is approaching positive infinity. Okay, we can zoom out and you can see the more and more you move to the right as X approaches positive infinity are the function is getting higher and higher. So as X approaches positive infinity, our function approaches positive infinity, and that's why our function has an infinite limit. The limit of our function as X approaches infinity is infinity.

This is problem number thirty four of the Stewart character. The Safe Edition Section two point six. Find the limit or show, but it does not exist. The Ltd's expertise. Negative infinity. I want those extra sixth over X to the force of plus one s o. One thing Khun do is we can make a choice defending the numerator and the denominator by exit the fourth and in this case, choosing anything greater than except forthe, such as Exit six. Ah, well, not benefit us since it will make the dominator a purse era. So the choice effects of the fourth should work well. Numerator becomes one of Rex to the fourth plys X squared. Since that is extra sixty minute makes in the fourth continuing to do any extra forth and the new denominator yet one plus one of Rex of the Force Ah, as we know a pro as we approach negative infinity, Each of these terms won a race for the fourth vanishes The approach Ciro and what were left over with is X squared over one. And as we approach negative infinity X squared approaches positive infinity. So this functioned averages does not the limit is that exists as thie limited approaching positive infinity as experts native

A problem. Number 49. Assume that X is positive support that excellent. It's bigger than you 21 over X Weird's one over X s words minus zero in smaller than abstinence. So one over X squared in a smaller than excellent Ah, if X squared is its positives to we can say that one over x square. It's a smaller than expected. So from here we can see that X is bigger than a square. Root off one over absence So this can be and eso where the definition Then it give us that the limit of extents infinity, one over X squared it's equal to you were in is equal to the square root off one over.

Problem number 50. Assume that X is its bothered them. Ah, and support that abstinence is bigger than zero. So absolute one over X A plus two minus zero is smarter than excellent. UH, which is equivalent to went over Exit plus two in the smaller than extra. Were experts to his positive, so X plus through it's bigger than went over. Excellent. So X is bigger than one who perhaps been minus the once you can be and then so but the definition. 1.4 point two the limit or extends to positive infinity off one over X plus through is equal to you know where n is equal to one over X minus two.