Question

For the given function f(x), the point c, and a positive number ε, find L=lim x→c

f(x). Then find a number δ>0 such that for all x,0<∣x−c∣<δ→∣f(x)−L∣<ε. f(x)= 1−4x
,c=−2,ε=0.5 L= (Simplify your answer.) For the given function f(x) and values of L,C, and ε>0 find the largest open interval about c on which the inequality ∣f(x)−L∣<ε holds. Then determine the largest value for δ>0 such that 0<∣x−c∣<δ→∣f(x)−L∣<ε. f(x)= 9x+76
,L=11,c=5,ε=0.09 The largest open interval about c on which the inequality ∣f(x)−L∣<ε holds is (Use interval notation. Round to four decimal places as needed.)

Answer 1

Let's start with the first function f(x) = 1 - 4x. The point we're returning to, c, is -2, and the provided epsilon, ε, is 0.5.

1. First, find L, which is the limit as x approaches c. So, put c into f(x):
f(c) = 1 - 4*(-2) = 9
So, L = 9.

2. Now find δ such that whenever the distance between x and c is less than δ (0<∣x−c∣<δ), the distance between f(x) and L is less than ε (∣f(x)−L∣<ε). We can find δ through the following equation:
δ = | ((L - ε) / 4) |
When the values are applied, we get:
δ = | ((9 - 0.5) / 4) | = 2.125

Now let's move on to the second function: f(x) = 9x + 76, where L=11, c=5, ε=0.09.

1. Here, L value is given, L = 11.

2. The second task is to find the largest value for δ such that whenever the distance between x and c is less than δ (0<|x-c|<δ), the distance between f(x) and L is less than ε (|f(x)-L|<ε). To achieve this, we first find the range of x such that the inequality |f(x) - L| < ε holds. Solve for x in the equation |f(x) - L| = ε we get two results for x (x_1, x_2).

3. The open interval is (x_1, x_2). The maximum δ will be the maximum of |x - c| for x in this interval. Unfortunately, without the specific values for the roots x_1 and x_2, we can't determine the exact values of the interval or δ.

However, in Python you would generally use the numpy root function (np.roots) to calculate the roots of equations, or the values of x that make the equation zero. Once you have the roots you can find the interval (x_1, x_2) and the maximum δ.