lim⁡𝑥→∞12+𝑥−3𝑥2𝑥2−4=x→∞lim​ x 2 −412+x−3x 2 ​ =

To solve this limit, we can use the rule of L'Hopital, which states that the limit of a quotient of two functions as x approaches a certain value is equal to the limit of the quotients of their derivatives.

First, let's simplify the expression:

lim (x→∞) (12+x−3x²) / (x²−4)

This can be rewritten as:

lim (x→∞) (3x² + x - 12) / (x² - 4)

Now, let's find the derivatives of the numerator and the denominator:

Derivative of the numerator: 6x + 1 Derivative of the denominator: 2x

Now, we can apply L'Hopital's rule:

lim (x→∞) (6x + 1) / (2x)

This simplifies to:

lim (x→∞) 3 + 1/(2x)

As x approaches infinity, 1/(2x) approaches 0. Therefore, the limit is 3.