(A) (e¹⁰ − 1) / [2e²(e² − 1)]

(B) (e²⁰ − 1) / [2e²(e² − 1)]

(C) (e¹⁰ − 1) / [2(e² − 1)]

(D) (e²⁰ − 1) / [2(e² − 1)]

Correct Answer: (e²⁰ − 1) / [2(e² − 1)]

Explanation

We first simplify the numerator using logarithmic properties.

logₑ(sec(ex) · sec(e²x) · … · sec(e¹⁰x))

Using log(ab) = log a + log b:

= Σ logₑ(sec(eᵏx)), where k = 1 to 10

For small values of t,

sec t ≈ 1 + t²/2

Taking logarithm:

logₑ(sec t) ≈ t²/2

Now substitute t = eᵏx:

logₑ(sec(eᵏx)) ≈ (e²ᵏ x²)/2

Hence numerator becomes:

Σ (e²ᵏ x²)/2 = (x²/2) Σ e²ᵏ

This is a geometric series:

Σ e²ᵏ = e² (e²⁰ − 1)/(e² − 1)

So numerator ≈

(x²/2) · e² (e²⁰ − 1)/(e² − 1)

Now simplify the denominator:

e² − e^{2cos x}

Using cos x ≈ 1 − x²/2:

2cos x ≈ 2 − x²

So,

e^{2cos x} ≈ e² · e^−x² ≈ e²(1 − x²)

Therefore,

e² − e²(1 − x²) = e²x²

Now take the limit:

Limit = [ (x²/2) · e² (e²⁰ − 1)/(e² − 1) ] / (e²x²)

Cancel x² and e²:

= (e²⁰ − 1) / [2(e² − 1)]

Hence, the required value is

(e²⁰ − 1) / [2(e² − 1)]

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