The smallest positive root of the equation tan x = x lies in

The smallest positive root of the equation tan x = x lies in the interval (0, π/2).

Here's the step-by-step solution:

1. The function tan x is periodic with period π, and it is increasing on each of its periods. Therefore, the equation tan x = x has at most one root in each interval of the form (nπ, (n+1)π), where n is an integer.

2. The function y = x is a straight line passing through the origin, and it intersects the y-axis at the point (0,0).

3. The function y = tan x also passes through the origin, but it approaches negative infinity as x approaches -π/2 from the right, and it approaches positive infinity as x approaches π/2 from the left.

4. Therefore, the two functions y = x and y = tan x intersect exactly once in the interval (0, π/2). This is the smallest positive root of the equation tan x = x.

5. To find the exact value of this root, we would need to use a numerical method, such as the bisection method or Newton's method, because the equation tan x = x cannot be solved exactly using elementary functions.