Find the limit.limx→4(x2−6x+8x2−16)lim𝑥→4⁡(𝑥2−6𝑥+8𝑥2−16)

The given expression is:

lim (x→4) [(x² - 6x + 8) / (x² - 16)]

First, simplify the expression:

= lim (x→4) [(x - 2)² / (x - 4)(x + 4)]

Next, factor out the numerator and denominator:

= lim (x→4) [(x - 2)(x - 2) / (x - 4)(x + 4)]

Now, we can see that as x approaches 4, the denominator approaches 0, which would normally indicate a point of discontinuity. However, the numerator also approaches 0, which suggests that we may be dealing with an indeterminate form of type 0/0.

In such cases, we can apply L'Hopital's rule, which states that the limit of a quotient of two functions that both approach 0 or both approach infinity is equal to the limit of the quotients of their derivatives.

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

Numerator derivative: 2(x - 2) Denominator derivative: 2x

Now, apply L'Hopital's rule:

= lim (x→4) [(2(x - 2)) / (2x)]

Simplify the expression:

= lim (x→4) [(x - 2) / x]

Finally, substitute x = 4 into the simplified expression:

= (4 - 2) / 4 = 2 / 4 = 0.5

So, the limit of the given expression as x approaches 4 is 0.5.