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Numerade Educator



Problem 65 Hard Difficulty

Prove that cosine is a continuous function.


$\begin{aligned} \lim _{h \rightarrow 0} \cos (a+h) &=\lim _{h \rightarrow 0}(\cos a \cos h-\sin a \sin h)=\lim _{h \rightarrow 0}(\cos a \cos h)-\lim _{h \rightarrow 0}(\sin a \sin h) \\ &=\left(\lim _{h \rightarrow 0} \cos a\right)\left(\lim _{h \rightarrow 0} \cos h\right)-\left(\lim _{h \rightarrow 0} \sin a\right)\left(\lim _{h \rightarrow 0} \sin h\right)=(\cos a)(1)-(\sin a)(0)=\cos a \end{aligned}$


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Video Transcript

This is a problem. Number sixty five of the sewer calculus eighth edition section two point five prove that coastline is a continuous function and we have shown on this previously that a continuous function is ah on Lee continues if and only if the limit as experts saying for the function in this case coastline. Well, I'll just write it in general, as of X is equal to today. Another way that we have seen this is the limit is X approaches or, as H approaches, zero out of the function. Ah h plus or April's age is equal test today and this is the approaching will be taking over the coast and function on the essentially the limit as H approaches. Ah, zero. Of course I kn April's age. We want to confirm that this is equal, indeed equal to cosign away. And if so, then we have proven that this function causing is continuous. What we're going to do is use a an identity for concerning that states. Cozy enough, X plus y is equal to consent of X cosign of wine minus scientifics Sign wine. We're going to confirm this exactly here and work with this apart. This will be limited approaches zero. Of course, I kn eh, Because an age minus And since it's a different part we use is Lim long to separate the limit. A new limit. Each man h goes zero for the sign terms Sign of a sign of age for the second limit as h Parcheesi room sign of age approaches zero because st zero is equal to zero. Um, so this second limit ends up that machine. Our first limit can be separate into two limits. He's in one of our limit lines. The limit is H project there. Of course, standing at members age approaches here. Of course, I'm h co sign approaches one as a chopper to zero cousin of H. Purchase one is H purchase zero. So we're left with the first limit, which is the limit is H zero of co sanity. And since cousin raise independent of age, then this limit is just equal to consider A And what we've done is confirmed that this initial limit is equal to Cazenove, which was our intent, and thus we can confirm that the cosan function is continuous