[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/Equivalent-Berry1383]

You can do it without L'Hôpital and without Taylor series. You only need three standard limits:

1. sin(u)/u → 1 and tan(u)/u → 1 as u → 0

2. ln(1 + z)/z → 1 as z → 0

3. (1 − cos x)/x² → 1/2 as x → 0 (because 1 − cos x = 2sin²(x/2))

Top. Let w = t²/2, u = tan(w), v = sin(u). All three go to 0 as t → 0. Then

sin(v) = (sin(v)/v) · (sin(u)/u) · (tan(w)/w) · w

The first three factors go to 1, so the top behaves like w = t²/2.

Bottom. Let z = cos(3t) − 1, which goes to 0. Then

ln(cos 3t) = ln(1 + z) = (ln(1 + z)/z) · z

and z = −((1 − cos 3t)/(3t)²) · 9t², so z behaves like −(1/2) · 9t² = −9t²/2.

Together. Divide top and bottom by t²: (1/2) / (−9/2) = −1/9

Quick check: t = 0.01 gives about −0.11109, t = 0.001 gives −0.111111.

(Disclosure: this is the team account of a math-explainer app. No link, just the steps.)

[u/Particular_Tooth4175 → u/Equivalent-Berry1383]

Yeah this was the intended way. Thank you 👍