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University of North Texas

# For what values of $a$ and $b$ is the following equation true?$$\displaystyle \lim_{x\to 0} \left( \frac{\sin 2^x}{x^3} + a + \frac{b}{x^2} \right) = 0$$

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we want to find values of A and B for the following limit as experts zero. That makes it equal to zero. So if we want this to be true, well, first, let's go ahead and just simplify this a little bit. So the first thing we can do is distributed across the pluses, and really, I'll just go ahead and group it like this to start. So the limit as X approaches zero, uh, sign of two x over execute. And actually, I'll go ahead and combined these two just by adding them really quickly. So in the new mirror, that's gonna be a plus B X all over, and then we'll have the x cute. So we'll have that and will have plus the limit as ex approach zero of a 0 to 0. And we know that a is a constant with respect to X. So this year will just end up being a so we can go ahead and some truck that over and so will get negative. A is equal to the limit as X approaches zero of signed to X plus bx over x cute. So we have this now. Now, if we were to just plug this in directly. Well, we're going to have zero. They're zero. They're enduring the denominator. So we have zero over zero, which means we can imply low petals rule to this. So for this next up, we're going to apply Loki tolls. And so that's going to give us. That's the limit. As X approaches zero of what the derivative of sign is co sign to X, and then we have to apply chain rules that we take the drone on the inside. So you multiply that by two, and then we take the derivative of B X, which just leaves us with B and then in the denominator will have three x squared. And this year should still be equal to negative, eh? Okay, Now, if we were to apply the limit directly, so co sign of zero is one. So, Wendell, with two plus b and then the denominator, we end up with zero. So we still can't divide by zero. But we want this limit toe hold. So that means So this implies we want our numerator toe also be zero. So we're gonna want to plus B to be equal to zero, which implies B is equal to negative too. So we found one value. So I'm just gonna go ahead and put that up here at the start. So B is equal to two, and now we can go ahead and plug that in for B. And doing that will give us something that we can take the derivative with respect to Earth. Apply low petals rule. Since, what? Zero over zero. So I'll just go ahead and apply Loki tall to walk in. So that's going to be the limit as X approaches zero of So we're going to have too. So the derivative of co sign is going to be negative. Sign to X, and then we have that, too, on inside that we have to take the derivative look that out front. So this too would actually become a four. And then we divide this by so we'd get X. And then we have to move that too out front by power room. And so then we'd get two times three or six. Okay. No, If we were to apply the lead here again, Sign of zero is gonna be zero X is just going to be zero. So we can imply. Loki told you one more time and I'll go ahead and move this up here. So by Loki tolls rule again. And actually, one thing I should still have over here is that the right hand side of our equation should still be equal to negative eight. Yeah. So now, using low petals ruled, the left hand side is going to become the limit as Ex purchase zero. Oh, so we take the derivative of the numerator again. So the derivative of sign is going to be co side to X. Then we have to take the derivative of change rule. So we end up with another two, multiply that by the negative for we already had that becomes negative eight and then in the denominator, the derivative of six X, which is be six. And this here is still equal to negative. Now we can go ahead and applied the limit again and again. Co sign of zero is going to be one. So we'll end up with negative eight over six. Busy, eh? Are negative, eh? So we could multiply the negative over and simplify it over six to give us 4/3. So we end up with that. A is equal to 4/3 and B is equal to negative too.

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