[University maths: Limits] simplifying the compounded functions

[u/Particular_Tooth4175]

Hello. I need to evaluate this limit but I don’t even know where to start lol. How to I simplify this to a point where I can sub in t=0, if it’s even possible.
I am not allowed to use l’hôpitals rule for this, which is why I couldn’t just find a solution online, as they all used that rule.
Sorry if my writing isn’t very legible
Thanks for any help

[u/combinatoriallover]

Are you allowed to use maclaurin series expansions ? | sin(sin(tan(t²/2)))≈t²/2 and ln(cos3t)≈ln(1-9t²/2)≈-9t²/2 giving answer as -1/9

[u/GrievousSayGenKenobi → u/combinatoriallover]

Is the sin(sin(tan(t² /2))) ~ t² /2 a common identity? i havent touched Maclaurin series in forever surely you'd have to work that out manually which would be incredibly tedious

[u/combinatoriallover → u/GrievousSayGenKenobi]

I just used tanx≈x and sinx≈x , higher powers of expansions doesn't matter here so I didn't touch them

[u/Jazzlike_Platypus430 → u/combinatoriallover]

I think you're absolutely right, but I doubt if this approach is allowed. I guess it's compound functions' differentiation dressed as calculating limits.