I'm trying to factor this equation in order to find its limit, but I'm not sure how to go about it. I can get the answer by graphing on a calculator but I need to do it without one.

Function:
F(x)=(x^2-36)/(sqrt(x^2-12x+36))
left, right, and overall lim as it approaches 6

let's look at that bottom part

√(x^2 - 12x + 36)
= √(x-6)^2
= x-6

so you have
f(x) = (x-6)(x+6)/x-6)
= x+6

so as x --> 6
limit f(x) = 12

To find the limit of the given function as it approaches 6, you can start by factoring the denominator and then simplifying the expression. Here's the step-by-step process:

1. First, let's factor the quadratic term in the denominator, x^2 - 12x + 36. This quadratic can be factored as (x - 6)^2, which is a perfect square.

2. Now, the function can be rewritten as F(x) = (x^2 - 36) / sqrt((x - 6)^2).

3. Next, simplify the expression inside the square root. Since we have a perfect square in the denominator, the square root of (x - 6)^2 simplifies to |x - 6|.

4. The expression now becomes F(x) = (x^2 - 36) / |x - 6|.

5. To find the left-hand limit as x approaches 6, you need to evaluate F(x) as x approaches 6 from the left side. This means x values that are smaller than 6.

6. Consider x = 5.99, which is slightly less than 6. Substituting this value into the expression gives F(5.99) = (5.99^2 - 36) / |5.99 - 6|.

7. Evaluating the numerator and denominator gives F(5.99) = (-15.1201) / (0.01) = -1512.01.

8. Repeat this process with x values that are progressively closer to 6 from the left side (e.g., 5.999, 5.9999, etc.) to verify the limit.

9. To find the right-hand limit as x approaches 6, repeat steps 5 to 8, but evaluate F(x) as x approaches 6 from the right side. This means x values that are greater than 6.

10. Lastly, to find the overall limit as x approaches 6, compare the left-hand and right-hand limits. If they are equal, then the overall limit exists and is equal to those values. If they are not equal, then the limit does not exist.

By following these steps, you can find the left, right, and overall limits of the given function as it approaches 6 without using a calculator.