Proving a Limit Involving Trigonometric Functions Using Taylor Series

In this article, we will delve into the process of proving a limit involving trigonometric functions, specifically the case where we need to show that the following limit is equal to (frac{1}{16}).

(lim_{x to frac{pi}{2}} frac{cot x - cos x}{pi - 2x^3} frac{1}{16})

Step 1: Taylor Series Expansion

To approach this limit problem, we will use the Taylor series expansions for (cot x) and (cos x) around the point (x frac{pi}{2}).

Cotangent Function

The cotangent function, (cot x), can be rewritten as:

(cot x frac{cos x}{sin x})

Near (x frac{pi}{2}), we have: (-cos x approx 0) (sin x approx 1)

Thus, (cot x) behaves similarly to (cos x) near (frac{pi}{2}). Using the Taylor series expansion for (cos x) around (x frac{pi}{2}), we get:

(cos x -x - frac{pi}{2} O(x - frac{pi}{2})^3)

For (sin x), the Taylor series is:

(sin x 1 - frac{(x - frac{pi}{2})^2}{2} O(x - frac{pi}{2})^4)

Thus:

(cot x frac{-x - frac{pi}{2} O(x - frac{pi}{2})^3}{1 - frac{(x - frac{pi}{2})^2}{2} O(x - frac{pi}{2})^4})

Using the first-order approximation for the denominator, we get:

(cot x approx - x - frac{pi}{2} - frac{(x - frac{pi}{2})^3}{2} O(x - frac{pi}{2})^3 approx - x - frac{pi}{2} - frac{1}{2}(x - frac{pi}{2})^3)

Combining the Two Series

Substituting these results into (cot x - cos x), we get:

(cot x - cos x left(- x - frac{pi}{2} - frac{(x - frac{pi}{2})^3}{2}right) - left(- x - frac{pi}{2} O(x - frac{pi}{2})^3right))

The linear terms cancel, leaving us with:

(cot x - cos x approx -frac{(x - frac{pi}{2})^3}{2})

Step 2: Substitute into the Limit

Now, substituting this result into the original limit, we get:

(lim_{x to frac{pi}{2}} frac{cot x - cos x}{pi - 2x^3} lim_{x to frac{pi}{2}} frac{-frac{(x - frac{pi}{2})^3}{2}}{pi - 2x^3})

Step 3: Simplifying the Denominator

Notice that as (x to frac{pi}{2}), we have:

(pi - 2x pi - 2left(frac{pi}{2} - xright) -2x frac{pi}{2})

Thus, (pi - 2x^3 approx -8x frac{pi}{2}).

Step 4: Final Limit Calculation

Substituting this back into our limit, we get:

(lim_{x to frac{pi}{2}} frac{-frac{(x - frac{pi}{2})^3}{2}}{-8x frac{pi}{2}} lim_{x to frac{pi}{2}} frac{frac{1}{2}}{8} frac{1}{16})

Therefore, we conclude that:

(lim_{x to frac{pi}{2}} frac{cot x - cos x}{pi - 2x^3} frac{1}{16})

Conclusion

Thus, the limit is proven to be:

(boxed{frac{1}{16}})