lim x->(pi/3) (cos x -(1/2))/(x-(pi/3)) is?

Since the ratio becomes indeterminate (0/0) at x = pi/3, take the ratio of the derivatives of numerator and denominator at x = pi/3.

lim x->(pi/3) (-sin x)/(1)
= -sin (pi/3)
= -(sqrt3)/2 = -0.866..

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To evaluate the given limit, we will simplify the expression and then substitute the given value of x into the expression.

Let's start by simplifying the expression:

lim x->(pi/3) (cos x - 1/2)/(x - pi/3)

We can rewrite the numerator as a single fraction:

lim x->(pi/3) ((2cos x - 1)/(2))/(x - pi/3)

Now we have a fraction divided by (x - pi/3), which can be rewritten as the product of the numerator and the reciprocal of the denominator:

lim x->(pi/3) ((2cos x - 1)/(2)) * (1/(x - pi/3))

Next, let's factor out the constant 2 from the numerator:

lim x->(pi/3) (2(cos x - 1/2))/(2(x - pi/3))

Now we can cancel out the 2 in the numerator and denominator:

lim x->(pi/3) (cos x - 1/2)/(x - pi/3)

Finally, we can substitute x = pi/3 into the expression:

(cos(pi/3) - 1/2)/(pi/3 - pi/3)

cos(pi/3) equals 1/2, and pi/3 - pi/3 equals 0, so we have:

(1/2 - 1/2)/0

Since we have an indeterminate form of 0/0, we can use L'Hôpital's Rule to find the limit.

Differentiating the numerator and the denominator, we get:

lim x->(pi/3) -sin x/1

Now we can substitute x = pi/3 into the expression:

-sin(pi/3)/1

sin(pi/3) equals √3/2, so we have:

-√3/2

Therefore, the limit of lim x->(pi/3) (cos x - 1/2)/(x - pi/3) is -√3/2.