Final answer:
To solve the limit problem, substitute x = π/6 into the expression, simplify, and find that the limit is 2. The calculation involves basic trigonometric values at π/6 and algebraic simplification.
Step-by-step explanation:
We need to determine the limit of the expression 6x - π √(3)/sinx - cosx as x approaches π/6. To do this, let us plug in the value of x = π/6 directly into the expression and check if it yields an undefined form or a real number.
Substituting x = π/6, we obtain:
(6(π/6) - π √(3))/(sin(π/6) - cos(π/6))
= (π - π √(3))/(1/2 - √(3)/2)
Since the denominator is not zero, we can simplify this expression:
= (1 - √(3))/(1/2 - √(3)/2)
By multiplying the numerator and the denominator by 2 to clear the fraction, we get:
= 2(1 - √(3))/(1 - √(3))
Now, since the numerator and denominator are equal (ignoring the factor of 2), they cancel each other out, leaving us with:
= 2
Therefore, the limit of the expression as x approaches π/6 is 2.