HELP NEEDED. What is the result?

[Rotokolo]

I’m struggeling, 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 paranthesis 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).

 

[pka]

I’m struggeling, 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 paranthesis in this case.

It is just a difference in notation. $\displaystyle \sin^4(x)$ is the common notation for the fourth power of the sine function evaluated at $\displaystyle x$.
However, some authors will write that as $\displaystyle \sin(x)^4$ or more clearly as $\displaystyle (\sin(x))^4=[\sin(x)]^4=\{\sin(x)\}^4$. Just a difference in notation.

 

[Rotokolo]

It is just a difference in notation. $\displaystyle \sin^4(x)$ is the common notation for the fourth power of the sine function evaluated at $\displaystyle x$.
However, some authors will write that as $\displaystyle \sin(x)^4$ or more clearly as $\displaystyle (\sin(x))^4=[\sin(x)]^4=\{\sin(x)\}^4$. Just a difference in notation.

Thank you so much for your explanation! Appriciate it. You helped a lot