Please show all steps Evaluate lim SolutionSolution Factoris

Please show all steps.

Evaluate:

Solution

Solution:

Factorise NUMERATOR and DENOMINATOR seperately

then apply limits.

3x²-24x+36=x² -8x+12

=x² -6x-2x+12

=x(x-6)-2(x-6)

=(x-2)(x-6)

x³ +6x² -16x=x(x²+6x-16)

=x(x² +8x-2x-16)

=x[x(x+8)-2(x+8)

=x[(x+8)(x-2)]

therfore

Lim_x-->2( 3x²-24x+36)/(x³ +6x² -16x ) = (x-2)(x-6)/x[(x+8)(x-2)]

cancel (x-2) from numerator and denoinator

Lim_x-->2( 3x²-24x+36)/(x³ +6x² -16x )=(x-6)/x(x+8)

now apply limits x---->2

Lim_x-->2( 3x²-24x+36)/(x³ +6x² -16x )=(2-6)/2(2+8)

= -4/20

   =-1/5