How do you use the epsilon delta definition to prove that the limit of #sqrt(x+6)=9# as #x->3#?

I will think backwards

plus nine

^2

minus 9

#18 < (9 + epsilon)^2 + (9 - epsilon)^2 = 81 + 81 +(18-18)epsilon

• 2epsilon^2#

To use the epsilon-delta definition to prove that the limit of sqrt(x+6) is 9 as x approaches 3, we need to show that for any given epsilon greater than 0, there exists a delta greater than 0 such that if 0 < |x - 3| < delta, then |sqrt(x+6) - 9| < epsilon.

Let's proceed with the proof:

Given epsilon > 0, we need to find a suitable delta.

We start by manipulating the expression |sqrt(x+6) - 9| < epsilon:

|sqrt(x+6) - 9| < epsilon |sqrt(x+6) - 9| * |sqrt(x+6) + 9| < epsilon * |sqrt(x+6) + 9| |x+6 - 81| < epsilon * |sqrt(x+6) + 9| |x - 75| < epsilon * |sqrt(x+6) + 9|

Now, we can see that if we choose delta = epsilon, we can simplify the expression further:

|x - 75| < epsilon * |sqrt(x+6) + 9| |x - 3 + 72| < epsilon * |sqrt(x+6) + 9| |x - 3| + 72 < epsilon * |sqrt(x+6) + 9|

Since we want to prove that the limit of sqrt(x+6) is 9 as x approaches 3, we can assume that |x - 3| < 1 (or any other suitable value) to simplify the expression even further:

|x - 3| + 72 < epsilon * |sqrt(x+6) + 9| |x - 3| + 72 < epsilon * (sqrt(x+6) + 9) |x - 3| + 72 < epsilon * (sqrt(3+6) + 9) |x - 3| + 72 < epsilon * (sqrt(9) + 9) |x - 3| + 72 < epsilon * (3 + 9) |x - 3| + 72 < epsilon * 12

Now, we can see that if we choose delta = min(1, epsilon/12), we can simplify the expression further:

|x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 3| + 72 < epsilon * 12 |x - 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