Evaluate lim x→0 (1/x) − (1/tan x).

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Multiple Choice

Evaluate lim x→0 (1/x) − (1/tan x).

Explanation:
Small-angle behavior of tan x and how differences of reciprocals behave near zero. Start by rewriting the expression as a single fraction: (1/x) − (1/tan x) = (tan x − x) / (x tan x). For small x, tan x has the expansion tan x = x + x^3/3 + O(x^5). So tan x − x = x^3/3 + O(x^5). The denominator x tan x = x(x + x^3/3 + O(x^5)) = x^2 + O(x^4). Therefore the whole ratio is (x^3/3 + O(x^5)) / (x^2 + O(x^4)) = x/3 + O(x^3), which tends to 0 as x → 0. Thus the limit is 0.

Small-angle behavior of tan x and how differences of reciprocals behave near zero.

Start by rewriting the expression as a single fraction: (1/x) − (1/tan x) = (tan x − x) / (x tan x). For small x, tan x has the expansion tan x = x + x^3/3 + O(x^5). So tan x − x = x^3/3 + O(x^5). The denominator x tan x = x(x + x^3/3 + O(x^5)) = x^2 + O(x^4). Therefore the whole ratio is (x^3/3 + O(x^5)) / (x^2 + O(x^4)) = x/3 + O(x^3), which tends to 0 as x → 0.

Thus the limit is 0.

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