The discussion revolves around evaluating the limit of a trigonometric expression as \(x\) approaches 0, specifically \(\lim_{x \to 0} \frac{\tan(\cos(4x) - 1)}{3x \sin(\frac{4}{3} x)}\). The subject area includes trigonometric limits and properties of trigonometric functions near zero.
The discussion is active, with participants providing insights into the properties of trigonometric functions for small values of \(x\). Some guidance has been offered regarding the approximations that can be used in the limit evaluation, and there is a recognition of the relevance of these properties to the problem at hand.
Participants express uncertainty about the specific properties of trigonometric functions that apply to the limit problem, indicating a need for clarification on how these properties relate to the limit evaluation.
jing2178 said:What do you know about sin(x), cos(x) and tan(x) when x is very small?
songoku said:I am not sure what you mean, maybe like this:
a. when x is very small (close to zero):
the value of sin x is close to 0
the value of cos x is close to 1
the value of tan x is close to 0
or
b. when x is very small (close to zero):
sin x ≈ x
cos x ≈ 1 - 1/2 x2 ≈ 1
tan x ≈ x
but I still don't know what the properties related to the question
Mentallic said:You're looking to use the properties of b.
If \cos(x)\approx 1-x^2/2 then what is \cos(4x) approximately equal to?
What's \sin(4x/3) approximately equal to?
Finally, you'll need to also convert the tan function as well in the same fashion.