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Mixed Practice Evaluate the following limits using any appropriate techniquel 5) (z2 + 14_ lim 324) 100 51 | 322 + 15_ lim (5w2 10292038374943029173) W-DO 16_ lim ...

Question

Mixed Practice Evaluate the following limits using any appropriate techniquel 5) (z2 + 14_ lim 324) 100 51 | 322 + 15_ lim (5w2 10292038374943029173) W-DO 16_ lim (212o) t-00 T2 17. ling 215t - 3 18. lim (700 e7t19_ lim (In(4r)) -1 Fr_020. lim 1700 In(x

Mixed Practice Evaluate the following limits using any appropriate techniquel 5) (z2 + 14_ lim 324) 100 51 | 322 + 15_ lim (5w2 10292038374943029173) W-DO 16_ lim (212o) t-00 T2 17. ling 21 5t - 3 18. lim (700 e7t 19_ lim (In(4r)) -1 Fr_0 20. lim 1700 In(x



Answers

Limits Evaluate the following limits. Use l'Hópital's Rule when it is comvenient and applicable. $$\lim _{x \rightarrow-1} \frac{x^{3}-x^{2}-5 x-3}{x^{4}+2 x^{3}-x^{2}-4 x-2}$$

Okay, said notice that this form of coercion is appalling on you, so we can use our for no meal deal. That is the limits of our polynomial is equal to our polynomial evaluated at that point, just one. So let's plug in that point, you get to monastery close for five and then what is that? Put that into my calculator, and yet that's equal to it.

So we're told in the directions that plugging in the negative one would give the U. A. 0/0 and determine reform. So we're going to use the Beatles role derivative of the numerator would be three X squared minus two X minus five. Denominator four x. to the third plus six X squared minus two X minus four. So if we plug in a negative one we would have three plus two minus five. For the Numerator 5 -50. And for the denominator we'd have negative four plus six plus two minus four. And that would also be zebra. That's an indeterminate form. And we do like the tall again To the numerator. This time we'll have six X -2 denominator 12 X squared plus 12 X minus two. Alright, plugging in a negative one will have negative six minus two or negative eight.

Were given this rational function, and our job is to determine what happens. Teoh as X becomes extremely large in the positive direction as X approaches infinity, we'll go ahead and re express thes polynomial ills by factoring out the X cubed So 14 X cubed, divided by X Cubed is 14 three x squared, divided by X Cubed is three over X and two X divided by X. Cubed is two over X squared in the denominator. We're going to do the same thing we're gonna factor out this x cubed. We get 21 x squared over excuse is one over x two x over X cubed, just two over X squared and one over text cubes. Well, let's take a look. What happens as X approaches infinity X squared approaches infinity that could approaches it even faster and X cubed approaches infinity even faster. So here, thes five fractions are all going to approach zero. As their denominators get larger and larger and larger, the values of these fractions right there all gonna approach zero, so that tells us we can actually ignore them. They're not gonna contribute anything much to the function as X gets very large. So for large X, our function is going to pretty much look like or is going to behave just like the function. 14 x cubed over 21 x cute. We can go ahead and cancel out those X cubes. Do that now goodbye and end up with 14/21 which can be reduced even further. Both numerator and denominator can be divided by seven, so this fraction reduces to 2/3 the limit. As acts approaches positive infinity of our function app of axe is equal to 2/3.

Okay, This question would like us to evaluate the following limit. And if we look here, we see that we have a 1/3 outside this entire function. So what we could do, since it's not affected by X in any way, means that we can do the limit first, then apply the power. And now that we've got the power out side of the limit, weaken split it up once more in the numerator and the denominator, and then we'll still apply that power at the end. So now all we have to do is plug in. So we get zero minus zero, plus 3000 divided by zero minus one and raise that whole thing to the 1/3. For we just have que brew of negative 3000. And the negative doesn't really matter here because it's an odd route. And I'll just rewrite this as the key brute of 10 cute times three because 10 cubed is 1000 or sorry times negative one. Or this is just negative. 10 times the Q Bert of three and all right, that out a little better. And that's our final


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