#lim_(xrarrpi)(cosx-sin((3x)/2))/(cosx + (sinx)/2)#
#lim_(xrarrpi)(cosx-sin((3x)/2))/(cosx + (sinx)/2) =(cospi-sin((3pi)/2))/(cospi + (sinpi)/2) #
#= (-1 - sin (66/14))/(-1 + 1)#
#= (-1+1)/(-1+1)#
#= 0/0#
Apply L'Hopital's rule
#lim_(xrarra)f(x)/g(x) = lim_(xrarra)(f'(x))/(g'(x))#
#f(x) = cosx-sin((3x)/2)#
#g(x) = cosx + (sinx)/2#
Taking derivative of numerator
#d/dx(cosx-sin((3x)/2)) = d/dx(cos(x)) - d/dx(sin((3x)/2))#
#d/dx(cos(x))= -sin(x) #
Apply the chain rule
#d/dx(sin((3x)/2)) #
#f= sin(u),u = (3x)/2#
#= \frac{d}{du}(\sin \ (u\ )\ )\frac{cancel(d)}{cancel(d)cancel(x)}\ (\frac{3cancel(x)}{2}\ )#
#= cos(u)3/2 = (cos((3x)/2)(3)/2)#
And bring back first derivation
#= -sin(x) - (cos((3x)/2)(3)/2)#
Taking derivative of denominator
#d/dx(cos(x) + sin((x)/2))#
#= d/dx(cos(x)) + d/dx(sin((x)/2))#
# d/dx(cos(x)) = -sin(x)#
Apply the chain rule
#f = u,u = x/2#
#d/dx(sin((x)/2)) = d/(du)(sin(u))cancel(d)/cancel(dx)(cancel(x)/2)#
#= cos(u)1/2#
#= cos((x)/2)1/2#
Bring back 1st derivation
#= -sin(x)+ cos((x)/2)1/2#
Looking both the numerator and denominator i can guess its again a zero , so L'Hopital's rule again.
Numerator
#d/dx(-sin(x) - cos((3x)/2)) = -d/dxsin(x) - d/dxcos((3x)/2)#
#-d/dxsin(x) = -cos(x)#
#d/dx (cos((3x)/2)(3)/2)) = 3/2(d)/(du)cos(u)d/dx((3x)/2) #
#= 3/2(-sin(u))3/2#
#= 3/2(-sin((3x)/2))3/2#
#= -9/4 sin((3x)/2)#
Numerator = #-cos(x)+9/4 sin((3x)/2)#
Denominator = #-sin(x)+ cos((x)/2)1/2#
#d/dx(-sin(x)+ cos((x)/2)1/2) = d/dx(-sin(x)) + d/dx(cos((x)/2)1/2)#
#= -cos(x) + 1/2d/(du)cos(u)d/dxx/2#
#= -cos(x) + 1/2(-sin(x/2))1/2#
#sin(x) = 1/(csc(x))#
#= -cos(x) + 1/(csc(x/2) *2 *2)#
#= -cos(x) + 1/(4csc(x/2))#
#= -cos(x) + (-sin(x/2))/4#
#= -cos(x) - (sin(x/2))/4#
Now plug in #pi# instead of #x#
Numerator
#-cos(x)+9/4 sin((3x)/2) = -cos(pi)+9/4sin((3pi)/2)#
#sin(x) = cos(pi/2-3pi/2)#
#= -cos(pi)+9/4 cos(pi/2-(3pi)/2)#
#= -cos(pi)+9/4 cos((cancel(-2)^(-1)pi)/cancel(2)) #
# -cos(pi)+9/4 cos(-pi)#
Remember cos(-x) = cos(x)
# = -cos(pi)+9/4 cos(pi)#
#= cos(pi)(-1 + 9/4)#
#= cos(pi)((-4 + 9)/4)#
#= -1 * 5/4#
#= -5/4#
Denominator
#-cos(x) - (sin(x/2))/4#
#= -cos(pi) - (sin(pi/2))/4#
# -1(-1)-1/4#
#(1*4-1)/4#
#= 3/4#
So #lim_(xrarrpi)(cosx-sin((3x)/2))/(cosx + (sinx)/2) = -5/4 *4/3#
#=- 5/3#
