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(8 pts) Evaluute the follorire Irali Iftre Ilmit cors To: erlzt sta"e thislim 22427-4 lin 1722-42x- lim _ #+r_23-4 Limn 3-228-4...

Question

(8 pts) Evaluute the follorire Irali Iftre Ilmit cors To: erlzt sta"e thislim 22427-4 lin 1722-42x- lim _ #+r_23-4 Limn 3-228-4

(8 pts) Evaluute the follorire Irali Iftre Ilmit cors To: erlzt sta"e this lim 224 27-4 lin 1722-4 2x- lim _ #+r_ 23-4 Limn 3-228-4



Answers

Use L'Hôpital's rule to find each limit. $$ \lim _{x \rightarrow 2} \frac{x^{2}+x-6}{x^{2}+6 x-16} $$

This question it is told that we have to find the limit at X approaches to for the function access to report for minus 16 divided by X cubed minus eight. We can see that if we substitute here too, then you get tourists to the bar four minus 16 divided by two cubed minus eight. That is zero by zero form. So due to this reason I am going to use here, I lost with a little, I lost little girl says that we have to differentiate numerator and denominator before substituting their limits. So and will do differentiate numerator and denominator before substituting their limits. So we can see that we get here limit X approaches to derivative of numerator biz four X Q and derivative denominated these three access were And after dividing by access where we get to your limit except approaches to four by three into X. And now we substitute here too. So we got here four by three and go to That can be written nice paid by three. So eight x 3 is the answer for this question here. OK thank you

The human problem, we know that we're going to be dealing with the limit as X approaches for Of two X, which is going to equal eight. So in this case You see if you look at the Function two X. And we consider what happens as X approaches for Let's say we have 4.1. Okay. and 3.9 as the boundaries, which is this result and A 0.2 boundary. So it's two times whatever is the boundary that we set in the X direction. The boundary in the Y direction. So we see is that we're going to have our delta value, for example is zero point one and epsilon okay. Is going to equal two times delta. So we see the adults that just needs to equal epsilon over to. Um with that qualification, we know that our function will remain bounded within the sequence Regardless of what we do. So if we chose 399 one, Now we see it gets a lot tighter, but the rationale still stays the same. We have that now it's going to be 0.01 for adult a value. Um But delta is still going to equal excellent over two. So that's how you make sense of this.

William problem? We want to use continuity to evaluate the element. So we know that in this case the arc tangent is going to be continuous everywhere. So what we have is that if we take x squared dark tangent of X square minus four over three X squared minus six X. We see that we can evaluate this as long as it's signed equal to zero, we can evaluate it. So in this case we see that we can plug in X equals two. So this is going to be um two squared minus four, which is just zero and this value right here will not be zero, it'll be 12 minus um 12 actually, so it would be undefined at this point. However, we can factor this out and see what we end up getting is the art tangent of two thirds. That's because we can factor this, I'm going to X plus two that will come out.

Hello We have to find the limit extends to do limit extents to execute, managed to exist square xQ. Managed to exist square plus four X monitored divided by X squared plus x minus six. So further we will take common access card from the first two times. So this will be x minus two. So limit extras too. So access card will be comments. This pre X minus two plus four is common. So ex minister divide by X plus three in two X minus two. Okay, so we can read the limit exchange two X squared plus four X. Managed to divide by X plus three into ex Minister. So this will be canceled out. So noble for the value extends to X squared plus four, divide by X plus today. Okay, so when access tends to do so this will be to school plus four upon two plus three. So this will be a cost of four plus four. Do it very fine. That is a cost to eight upon five. Okay, so this is the answer. I hope you understood. Thank you.


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