What is the equation of the line normal to # f(x)=(xsinx)/tanx# at # x=pi/3#?

Answer 1

Equation of normal is #(y-pi/6)=(2sqrt3)/(pi-sqrt3)xx(x-pi/3)#

As #f(x)=(xsinx)/tanx#, at #x=pi/3#, #f(x)=(pi/3xxsqrt3/2)/sqrt3=pi/6#
Hence,m we are seeking equation of normal at #(pi/3,pi/6)#
As slope of tangent is given by #f'(x)#, as #f(x)=(xsinx)/(sinx/cosx)=xcosx#
Hence #f'(x)=cosx-xsinx# and at #x=pi/3#, #f'(pi/3)=cos(pi/3)-pi/3xxsin(pi/3)=1/2-pi/3xxsqrt3/2#
= #1/2-pi/(2sqrt3)=(sqrt3-pi)/(2sqrt3)#
Hence slope of tangent is #(sqrt3-pi)/(2sqrt3)#
and that of normal would be #(-1)/((sqrt3-pi)/(2sqrt3))=(2sqrt3)/(pi-sqrt3)#

and equation of normal would be

#(y-pi/6)=(2sqrt3)/(pi-sqrt3)xx(x-pi/3)#
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Answer 2

The equation of the line normal to f(x) = (xsinx)/tanx at x = pi/3 is y = -sqrt(3)x + sqrt(3)/2.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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