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University of North Texas



Problem 3 Easy Difficulty

Find the limit.

$ \lim_{t\to 0} \left(e^{-3t} i + \frac{t^2}{\sin^2 t}\ j + \cos 2t\ k \right) $



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Video Transcript

So if we want to find this limit here, we can go ahead and apply pretty much all the same limit rules that we've learned for the one dimensional case. We could first distribute this across the pluses and pull out any constant. So this I, j and K or assuming are constants. So we converse. Rewrite this as the limit as T approaches zero of e to the negative three t times, and I'll just write I out here and I'll write. This is I hat, um, plus the limit as he approaches zero of tea and I'm actually going to rewrite this as t over sign T uh, squared. And then on the outside we have this multiplied by J hat and then, uh, co sign or the limit as T approaches zero of co sign of two t times K hat so we could just go ahead and apply the limit directly here, my plug in and zero. So that would just be one times I hat Plus, now, this limit might look a little bit familiar to you if we were to kind of rewrite this as the limit. As fate approaches zero of sine of t over T. We know this is equal toe one. And we know if we were to actually reciprocate this, um, we would end up getting the same thick. So this also implies that the limit s fate approaches. Zero of tea over sine of T is equal to one as well. So that means we can pull this inside. And this limit is just going to be one. So it would be one squared J hat plus and then we just apply the limit. Here's would be co sign of zero and co sign of zero is one B k hat. So this just ends up being I had plus jihad plus k hot. And then this here is going to be our limit.