Given lim x>-4 1/2x-3 and E=.001 Find S that goes with this E

To find S that goes with the given ε (E), we need to determine a value for δ (δ > 0) such that whenever |x - c| < δ for x > -4, we have |f(x) - S| < ε.

Let's start by rewriting the given function f(x):

f(x) = 1/(2x - 3)

Now, let's work on finding appropriate values for δ and S.

Step 1: Find the limit of f(x) as x approaches -4:
To find the limit, substitute the limiting value into the function:

lim x→-4 (1/(2x - 3))

Since we are looking for the limit from the right side (x > -4), substitute -4 into the function:

lim x→-4+ (1/(2x - 3)) = 1/(2*(-4) - 3) = 1/(-8 - 3) = 1/(-11) = -1/11

Therefore, the limit as x approaches -4 from the right side is -1/11.

Step 2: Set up the inequality for |f(x) - S| < ε:

|f(x) - S| < ε
|1/(2x - 3) - S| < ε

To make the inequality easier to work with, we can start by setting ε = 0.001:

|1/(2x - 3) - S| < 0.001

Step 3: Determine a value for δ that ensures |f(x) - S| < ε:

To find a suitable δ value, we manipulate the expression |1/(2x - 3) - S| < 0.001 algebraically:

|1/(2x - 3) - S| < 0.001

To simplify further, we can make some assumptions by considering the worst-case scenario. Since the function f(x) approaches -1/11 as x approaches -4 from the right side, we can assume that the difference between f(x) and S is at most 1/11:

|1/(2x - 3) - S| ≤ 1/11

Therefore, we can set δ to ensure that |1/(2x - 3) - S| ≤ 1/11 is satisfied. Let's assume δ = 0.001:

|1/(2x - 3) - S| ≤ 1/11
1/(2x - 3) - S ≤ 1/11
1/(2x - 3) ≤ S + 1/11

To satisfy the above inequality, we need to choose a value for S that makes the right-hand side of the inequality (S + 1/11) smaller than or equal to 0.001. Let's subtract 1/11 from both sides:

1/(2x - 3) - 1/11 ≤ S

So, we have found S that goes with ε = 0.001: S = 1/(2x - 3) - 1/11.

Keep in mind that this value of S may not be the only option. You can choose other values for S that satisfy the inequality depending on your specific requirements.