5

2x _ 1 f dx x2 1 -...

Question

2x _ 1 f dx x2 1 -

2x _ 1 f dx x2 1 -



Answers

$f(x)=(1+2 x)^{1 / x}$

Okay, We're gonna compute the limit of the function one over two as X approaches. Negative too. And we're gonna do this. Using the grass is a function which you can see in your textbooks. It's gonna redraw it Here. He's our axes label, our axes X. Why? And since the limit we care about is that negative too? I'm just no label negative two on the extraction. And as you can see in your textbooks, the sunken look, something like and you can check for yourself. That curve makes sense just by plugging in some point table. Okay? We can immediately see that, Linda depends on whether you're coming from the right side or from the left side. We actually compute both right handed and left. Let's first approach negative from the right side. Okay. On the graph, we see that as a yet your negative too. From the right, the valley of the function wise explodes towards infinity. So the graph tells us that this limit for the positive in unity and let's just do a quick sanity check. If X is approaching native to from the right point, that means is greater than negative too. So explosive is a small, positive number and one over exposed to very large. Okay, and now we need to approach from the left. Well, you screen didn't know the left sided love it, and again they function one of them. And again, let's look at the graph. If X is approaching negative from the left, we see that the value of something word negative. The graph tells us that this love it, it's negative. And before we want to do a quick sanity, check ill effects unspoken, negative to fromthe left that which means that he was a small positive number and one over at small negative number and one over a small negative number is a large.

So they're given funds in areas signing myself Yeah X minus one Yeah minus two, You can, yeah. Mhm.

Objective is to find out Yes Plus two equal here. So let's go ahead and use U substitution view equals two X plus one. Yeah. Take that derivatives we have two units equal to DX. Okay, so then plug in X values. We have X equals nature too. Then you equals negative one here And that x equals one. Contact you single to to so you have the integral for negative too. Yeah. Take it upon two of africa. Look at that, that's exactly what we had 18. So that's equivalent to what we had started with here. So it's 5/2. So basically this value of this integral here, it's going to be able to fire over to. We'll be out there. Yeah. So it's just through like substitution, what we're doing here. That's what these are equal.

The question is to deter mine, but the given function is even or orden in off these. So the given function is if it's equals to one minus off, one minus X squared, with the power minus. How so now taking f off negative X, which will be equal to one minus minus X. Where one by WHO? Which will be 4 to 1 minus excess where one by the as minus Texas granite cools to excess Where so from this we came to know that F off negative X is a false toe. FX, which is the condition for even function so that given function is you want function. Thank you.


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