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(1 point) Evaluatee 3+3 -1+ 3x3 2x6 lim X-0 9x9Hint: Using power series:...

Question

(1 point) Evaluatee 3+3 -1+ 3x3 2x6 lim X-0 9x9Hint: Using power series:

(1 point) Evaluate e 3+3 -1+ 3x3 2x6 lim X-0 9x9 Hint: Using power series:



Answers

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 0} \frac{(1-2 x)^{-1 / 2}-e^{x}}{8 x^{2}}$$

To compute this limit, we substitute eat your ex power with the first three terms of its, see risks expansion And their sickles limit as goes to zero one plus x. task Tax squared over two plus the higher order terms of X squared miners were Cossacks over X squared. So there see calls limits text squared over two plus the higher order terms of X squared over X square. This is the Ecos and limits a half plus, limits the higher order terms of X squared over Aps square According to the definition of this term Mikel, Hence the limited as.

My name were given the limits from last few series. To evaluate this limit, This is the limit. As X approaches infinity of x squared times. Eat the negative one over x squared minus one, two. You just plug in X equals infinity. Where you have an infinity times beard oil louis and tim with beard oil zero, which is a undefined indefinite form. So instead let's try to reward the expression. Using series, you have X squared times negative one plus, each of the negative one over X square. Well, using a series expansion, this is X squared times negative one Plus and this is one -1 over x squared plus. And this is the same as negative one over X squared squared, which is one of the extra the fourth. So is plus 1/2 times X to the fourth Uh -6. I'm sorry, 1/3 factorial or six times X, 2 to 6 and so on. And so this simplifies to get rid of the ones we have negative one plus 1/2 X squared minus 1/6 X to the fourth. And so on, 4th degree their skin is melted off. They're facing house fire. I'd be like throw up mm and therefore playing this back in. You see that the limit is equal to the limit as X approaches infinity, uh, negatives one plus 1/2 X squared minus one of the six X to the fourth and so on. All the terms, except for the first one, have an X factor in the denominator and so they go to zero. And so our limit is simply the first term negative one. I was like, I was under the impression that I was here. So II

Were given a limit and rescue series to evaluate this limit. This is the limit as x approaches zero of game E. To the x -1 Plus X over X squared to find this limit first, let's use series to rewrite the expression. This is the same as one over X squared times E to the x minus one plus X. And then using the taylor series expansion. This is one over X squared times one plus X plus X squared and so on. Yeah, sorry, X squared over two factorial -1 plus x. Which is in turn equal to well we have X squared over two factorial times. One of brecht's squared is just 1/2 plus X over three factorial plus X squared over four factorial and so on. Therefore The limit as X approaches zero of this expression is the limit. As X approaches zero for small enough X of one half plus X over three factorial plus X squared to reform factorial and so on. Mercedes terms all have an X. And so in the limit they're all zero. So the limit is simply one half. You're

22. We want to evaluate the following limit by using a tentative Siris are limit is the metals. So we x employees power one over X minus one when extends toe infinity. Since we cannot expand the tale of stories, we have to rewrite the limit tow avoid extends to infinity. Noticed that if we use substitution off T equals one over X, we get yep equals Zima. Therefore, our limit. Well, now look like this. The limit off. One over. Team of supply E power T minus one when t tends to zero. Notice that without rewriting this limit, we would get equal 0/0, which makes no sense. So we need to think of something else. So we can now try with the table. Siri's e poverty equals submission from K equals zero Children. Infinity off the our key over key factory. Rocky. Using this, we get damn it. One over tea come to apply. Be party minus one. He tends to zero. You can then refight this equation in tow. Something like this. Summation from K equals zero g. Bulky for Tulowitzki, minus one. Booth limit off one over. Team of supply submission from key equals zero till infinity off. G que about G minus one. Over. I told you. Off key minus one over tea. Spooky. Then we can further simplify this in tow. It means when he tends to zero off. Thi hour zero minus one over a total of zero plus t about one once. One over. Factories of one plus Steve. Our two minus one over the factory off to Yes. Lusty out. Three minus one over in the factory off three. And so on. Minus one over tea. Our limit cannot group something like this. Teeth tends to zero one over tea. Plus keep our Brazil with factorial off. One ste one over. Factorial two plus give our two over. Factorial three. And so on. Minus one over tea. Further simplification. Limit off. One over. Factorial of one plus t over factory of two. Plus keep our toe over. Sectorial off three and so on. Therefore, I can't say that our limit, of course. The one last 0/2 years to plus zero power to over factorial of three and so on, which is equal to one plus zero plus zero and so on, which is equal toe one. This is our final answer


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