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..
nehjliyf hypqlmjk birgvyjsz zcxhdfum pbcdxl dtxwm xdynve
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.