Find the equation of the line tangent to gx x3 9x2 8 at x

Find the equation of the line tangent to g(x) = x^3 - 9x^2 + 8 at x = 2. Evaluate the following limits: lim_x rightarrow 2 x^3 - 3x^2 + 2x/x63 + 7x^2 + 12x lim_x rightarrow -3 x^2 + 11x + 24/2x^2 - 2x - 24

Solution

lim x--> 2   ( x^3 - 3x^2 + 2x ) / ( x^3 + 7x^2 + 12x )

= 2^3 - 3(2^2) + 2(2) / (2^3 + 7(2^2) + 12(2) )

= 0 / 60

= 0

b) lim x --> -3   ( x^2 + 11x + 24 ) / ( 2x^2 - 2x - 24 )

= factoring numerator and denominator

lim x --> -3   ( x+8)( x+3) / 2(x+3)(x-4 )

lim x --> -3 (x+8) / 2(x-4)

plugging x = -3

(-3+8 ) / 2 ( -3 - 4) = 5 / -14

therefore , answer is   -5 / 14