I found this in the solutions part of my...

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I found this in the solutions part of my textbook. Why does this even work? There's no explanation at all.

How is bringing up the derivative of cos here even relevant to finding this limit?

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>>7857855

They recognized the limit as the derivative of cos(x) at x = 0. Pretty funny desu.

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>>7857855

because the limit is exactly, term by term, the derivative of cos at 0. that's nice

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>>7857872

But why would they write this shit like that? That's like writing that the limit equals 5^(0)-1. Illogical, and makes no sense at all, even though it's technically correct.

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>>7857855

didn't he use l'hopital?

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>>7857881

they're not saying this limit is equal to this other limit.

They're saying this:

Let f(x) = (cos(x)-1)/x

Clearly f(x) = (cos(x) - cos(0)) / (x-0)

So lim(f) is lim of either expression

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>>7857886

I thought the same

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He used l'hÃ´pital's rule

which you should not use since you don't know what the fuck u are doing.

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>>7857899

Are you OP?

L'hopital is a method to solve limits which are undetermined (you know, when you evaluate and you get 0/0 and stuff like that)

If you are just learning limits, then I would assume your teacher didn't tell you about this. That's because teachers at first expect you to solve limits without l'hopital, to make sure you learn.

>teachers hates this trick

>learn this easy way to solve limits.

Okay, so assuming yo have an undetermined function. differentiate the numerator and then differentiate the denominator BY SEPARATE. Don't use division rule.

d/dx (x) = 1

d/dx (cos(x)) = -sin (x)

your limit will be lim x->0 -sin(x)/1 = 0

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>>7857855

It's literally the definition of "the derivative of the cosine function at 0"

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L'hopital's Rule...

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>>7857855

>literally what is the fundamental theorem of calculus

https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus

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>>7857950

So, L'hoital's rule. Got it.

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>>7857950

yeah yeah I saw it now.

but it's the same, it's literally the same in this problem.

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>How is bringing up the derivative of cos here even relevant to finding this limit?

What do you mean "how is it relevant"?

He recognized that the limit equals the derivative and it enables him to compute the value he wants to compute.

Say he wouldn't "bring up the derivative here". Well then he'd have to argue somehow else what the limit is.

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>>7857975

no. fucking retard.

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>>7857855

The people who wrote the textbook are clever. They used a holistic approach rather than a formulaic one. This is probably.used to determine who is just copying the answers.

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