Help! Confusion about this limit. Please help

Jan 2020
4
0
Slovakia
I’m struggling, with the result of this limit. Especially cos(x)^3 . Can someone explain what does it mean? According to my professor, it means cos^3(x); she is basically saying that cos(x)^3 = cos^3(x). I don’t think that it’s correct assumption. Can someone explain conventions of using parentheses in this case? Thank you so much!!

NOTE: I know the calculus, I know how to use powered cos, sin etc. and I know that cos^3(x) is different from cos(x^3).
 

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greg1313

Forum Staff
Oct 2008
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The limit is equivalent to

\(\displaystyle \lim_{x\to0}\frac{1-\cos^3(x)}{3x^2}\)

Apply L'Hospital's rule twice. The limit is 1/2.
 
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I’m struggling, with the result of this limit. Especially cos(x)^3 . Can someone explain what does it mean? According to my professor, it means cos^3(x); she is basically saying that cos(x)^3 = cos^3(x). I don’t think that it’s correct assumption. Can someone explain conventions of using parentheses in this case? Thank you so much!!

NOTE: I know the calculus, I know how to use powered cos, sin etc. and I know that cos^3(x) is different from cos(x^3).
It's a convention. $\cos^3(x)$ means $(\cos(x))^3$.

The notation $\cos(x)^3$ in my opinion is ambiguous. Your teacher should not have used it. Of course they're the teacher so don't go saying, "Some guy on the Internet says you're wrong." But in fact, teacher's wrong.
 
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Jan 2020
4
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Slovakia
It's a convention. $\cos^3(x)$ means $(\cos(x))^3$.

The notation $\cos(x)^3$ in my opinion is ambiguous. Your teacher should not have used it. Of course they're the teacher so don't go saying, "Some guy on the Internet says you're wrong." But in fact, teacher's wrong.
Thank you for your answer. My thoughts on my final were exactly same as yours. I might go and see my prof during her office hours. Appreciate your help!!
 

greg1313

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Oct 2008
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There's two parts to $\cos(x)$. "cos" and it's argument, "x". If "^3" doesn't apply to the argument, what does it apply to? Ambiguous, perhaps. Intractable, no.
 

skeeter

Math Team
Jul 2011
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FYI, calculator syntax interprets $\cos(x)^3$ as $\cos^3{x}$ ...

cosine_cubed.png
 
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SDK

Sep 2016
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USA
It's a convention. $\cos^3(x)$ means $(\cos(x))^3$.

The notation $\cos(x)^3$ in my opinion is ambiguous. Your teacher should not have used it. Of course they're the teacher so don't go saying, "Some guy on the Internet says you're wrong." But in fact, teacher's wrong.
This is completely backwards. Writing $f(x)^3$ is completely standard and it means exactly what his/her teacher is saying it does. The notation $f^3(x)$ is almost never used since it is reserved for function iteration. In fact, the trig functions are the only examples I know of which break this convention and god only knows why they are written like this. But the fact remains that writing $\cos(x)^3$ is still quite standard and as unambiguous as it gets since I don't know how else you could interpret this notation. His/her teacher is definitely not wrong.
 
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\The notation $f^3(x)$ is almost never used
Long time since you've been in high school. That notation is universal for trig functions as you agree. I take your points but stand by what I wrote. Teacher 's notation was ambiguous.