Evaluate the following limit. If the answer is positive infinite, type \"I\"; if

Question

Evaluate the following limit. If the answer is positive infinite, type "I"; if negative infinite, type "N"; and if it does not exist, type "D". limit at infinity of ((12x^3-5x+1)/(3x^3+10x+1))^(1/2)

Explanation / Answer

I lim_(x->infinity) sqrt((12 x^3-5 x+1)/(3 x^3+10 x+1)) Using the power law, write lim_(x->infinity) sqrt((12 x^3-5 x+1)/(3 x^3+10 x+1)) as sqrt(lim_(x->infinity) (12 x^3-5 x+1)/(3 x^3+10 x+1)): = sqrt(lim_(x->infinity) (12 x^3-5 x+1)/(3 x^3+10 x+1)) Indeterminate form of type infinity/infinity. Using L'Hospital's rule we have, lim_(x- >infinity) (12 x^3-5 x+1)/(3 x^3+10 x+1) = lim_(x->infinity) (( d(12 x^3-5 x+1))/( dx))/(( d(3 x^3+10 x+1))/( dx)): = sqrt(lim_(x->infinity) (36 x^2-5)/(9 x^2+10)) Indeterminate form of type infinity/infinity. Using L'Hospital's rule we have, lim_(x->infinity) (36 x^2-5)/(9 x^2+10) = lim_(x->infinity) (( d(36 x^2-5))/( dx))/(( d(9 x^2+10))/( dx)): Answer: {The limit of a constant is the constant:, = 2}

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