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The limit evaluates to
.
Solución

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Como
Beyond the Answer
To evaluate the limit
we start by substituting
. This will help us investigate the behavior of
as
approaches
.
Since
, we need to evaluate
. Depending on the value of
, the angle
could either be in the first quadrant (where both sine and cosine are positive) or in the second quadrant (where cosine is negative, but sine is positive).
Let’s analyze
in both scenarios:
- If
is less than 2, then will be in the first quadrant, and is well-defined and positive. - If
, then approaches as approaches . - For
, as approaches , will be negative.
Next, we will utilize L’Hôpital’s rule, since both the numerator and the denominator approach 0 or
as
, which leads us to calculate derivatives.
Calculating
gives:
- Numerator: As
, . - Denominator:
We can then apply L’Hôpital’s Rule:
Find
and
.
This might simplify as L’Hôpital’s Rule is applied iteratively until we obtain a recognizable limit.
After proper substitution, simplifications, and limits analysis, we can conclude:
If
:
Assuming proper calculation shows limits converge, we can determine a limit for
being more sensitive due to
in the form
.
In conclusion, evaluating numerically both around
gives us a constant depending on the relationships in cotangent values concluding the limit at specific numerical constants or diverging values based on hexagonal properties and limits leading to custom trigonometric behavior at
intersecting. Thereby, the limit reaches:
Through appropriate evaluation, we conclude behavior variants from
versus
behavior and discover potential values accordingly leading from first principles trigonometric basis, thus:
This context helps in numerical limits analysis leading to confirmation-based results from trigonometric calculus.

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