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The limit evaluates to .

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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:
  1. If is less than 2, then will be in the first quadrant, and is well-defined and positive.
  2. If , then approaches as approaches .
  3. 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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