Limit as x approaches 6:

((60-6x- abs(x^2-10x))/(abs(x^2-100)-64)

To find the limit as x approaches 6 for the given function:

((60 - 6x - |x^2 - 10x|) / (|x^2 - 100| - 64))

We can compute the limit by evaluating the left-hand and right-hand limits separately and checking if they are equal.

First, let's find the left-hand limit:

As x approaches 6 from the left-hand side (x < 6), we substitute values slightly less than 6 into the expression to see how it behaves. Let's pick x = 5.9:

((60 - 6(5.9) - |(5.9)^2 - 10(5.9)|) / (|(5.9)^2 - 100| - 64))

Simplifying this, we get:

((60 - 35.4 - |29.21 - 59|) / |29.21 - 100| - 64))

Calculating further:

((60 - 35.4 - |-29.79|) / |-70.79| - 64))

((60 - 35.4 - 29.79) / 70.79 - 64))

(24.6 - 29.79) / 6.79

-5.19 / 6.79 ≈ -0.7658

Next, let's find the right-hand limit:

As x approaches 6 from the right-hand side (x > 6), we substitute values slightly greater than 6 into the expression. Let's take x = 6.1:

((60 - 6(6.1) - |(6.1)^2 - 10(6.1)|) / (|(6.1)^2 - 100| - 64))

Simplifying this, we get:

((60 - 36.6 - |36.21 - 61|) / |36.21 - 100| - 64))

((60 - 36.6 - |-24.79|) / |-63.79| - 64))

((60 - 36.6 - 24.79) / 63.79 - 64)

(23.4 - 24.79) / -0.21

-1.39 / -0.21 ≈ 6.619

Since the left-hand limit is approximately -0.7658 and the right-hand limit is approximately 6.619, we can see that the limits from both sides are not equal.

Therefore, the limit does not exist as x approaches 6.