I found this in the solutions part of my textbook. Why does this

>>7857855
They recognized the limit as the derivative of cos(x) at x = 0. Pretty funny desu.

>>7857855
because the limit is exactly, term by term, the derivative of cos at 0. that's nice

>>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.

>>7857855
didn't he use l'hopital?

>>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

>>7857886
I thought the same

>>7857886
>>7857895
what the fuck are you talking about!?

He used l'hôpital's rule
which you should not use since you don't know what the fuck u are doing.

>>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

>>7857855
It's literally the definition of "the derivative of the cosine function at 0"

L'hopital's Rule...

>>7857912
>>7857916
>>7857927
he did not use l'hop you dense faggots. it's the fucking definition of the derivative at 0

>>7857855
>literally what is the fundamental theorem of calculus
https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus

>>7857950
So, L'hoital's rule. Got it.

>>7857950
yeah yeah I saw it now.
but it's the same, it's literally the same in this problem.

>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.

>>7857975
no. fucking retard.

>>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.