Question:

<p><strong>Step 1:</strong> Simplify the base. Note that tan(x/x) = tan(1), which is a positive constant ≈ 1.557.</p><p><strong>Step 2:</strong> Rewrite the limit: lim(x→0) [tan(1)]^(1/x). Here tan(1) is a fixed positive number and 1/x → ±∞ as x → 0.</p><p><strong>Step 3:</strong> Since tan(1) ≈ 1.557 > 1, we have [tan(1)]^(1/x) → ∞ as x → 0⁺ and [tan(1)]^(1/x) → 0 as x → 0⁻.</p><p><strong>Step 4:</strong> The left and right limits are different. However, if the question intends tan(x/x) as written literally (not tan(x)/x), and assuming the limit is requested as x → 0 through positive values only, or there is a typo in the original problem where it should be tan(x)/x or similar, the answer would be reconsidered.</p><p><strong>Clarification:</strong> If the question is actually lim(x→0) (tan x / x)^(1/x), then tan(x)/x → 1, giving 1^(∞). Using logarithms: ln L = (1/x)·ln(tan x / x) = (1/x)·[ln(tan x) - ln(x)]. By L'Hôpital or Taylor expansion, this → 0, so L = e⁰ = 1.</p><p>∴ Answer: <strong>1</strong></p>