5

$$lim _{x ightarrow 0} frac{x}{an ^{-1} 2 x} cdotleft{ext { Ans} frac{1}{2}ight}$$...

Question

$$lim _{x ightarrow 0} frac{x}{an ^{-1} 2 x} cdotleft{ext { Ans} frac{1}{2}ight}$$

$$ lim _{x ightarrow 0} frac{x}{ an ^{-1} 2 x} cdotleft{ ext { Ans, } frac{1}{2} ight} $$



Answers

$$ \lim _{x \rightarrow 0} \frac{x}{\tan ^{-1} 2 x} \cdot\left\{\text { Ans, } \frac{1}{2}\right\} $$

In this trouble whereas to find given limit. So we're gonna start a solution by defining the Newberry. What? Sign that you is people, too. Nature law. So as X approaches zero from positive side. Um, actually logged zero. We'll go to infinity, so promise you are 40 b positive so we can rewrite this limit as limited as Yugoslav affinity engine ewers. Oh, you and the use Grecian eight in chapter 3.7. We see that this limit is deep of finance. Fine over too.

If we were to plug in zero, we'd end up with 0/0. So we know that we must do something else or rearrange this in some way. One thing that we could do is rewrite tangent of two X as sign of two X, divided by co sign of two acts. And then, of course, we still have this divided by X here. And so we'll go ahead and also, um, have this division by X. I'm gonna go ahead and re arrange that just for a moment, though, and lead some space because in order for the sign appear to cancel, we would really want That's to be divided by two acts. So sign of two extradited by two X would all be one. However, we need to multiply by two over to to get this to to appear in the first place. And so now, substituting zero for this ex here again, we've taken care of all of this so you can ignore that stuff there. Um, we've now divided the sign to expect to x that gate that give us one. And then since the limit as X approaches zero and then substituting zero here code signing zero is also going to give us one and so really, we've now canceled everything, except that's too and so this limit we'll evaluate to a value of two.

One method solved this limit because if we use direct substitution and been zero, we'd get zero of top divided by zero. And so we could actually delight each of these terms by two X. That's doing so. Nobody read. Write this so this would be divided by two x, divided by two X and divided by two X. And then we'll get some cancellations to happen. So one of these exes will cancel with this and then substituting in will get 0/2, which would give us zero. The's exes would cancel, and that would leave us with negative 1/2 for the second term and then the third term, using the identity sign of X over X. As long as X is approaching, zero is going to end up giving us one. And so we have plus 1/2 at the end and therefore this limit We'll evaluate NBN 20

In this question. We need to find limit extents toe. I bet you Linus Dan next, not this within. My gut limit extends to my big toe minus sign. Next. My God, Now we know Limit Express toe Alberto minus. Sign knicks according to one and limit extra. Unstow. I like to minus close six. You know what's up? Your being here together will help. One way to go. It took less infinity.


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