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Determine the limit of the expression 6x− π sqrt(3)/sinx−cosx as x approaches π/6. Show your step-by-step calculations.

User Rrbest
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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.

User Oruen
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