Question

For the limit lim x → 2 (x3 − 4x + 9) = 9 illustrate the definition by finding the largest possible values of δ that correspond to ε = 0.2 and ε = 0.1

Answer 1

The question is about finding the values of δ corresponding to ε = 0.2 and 0.1 for the given limit. The process involves solving the inequality |x3 − 4x| < ε and finding the range for x that satisfies the condition, hence determining δ.

Step-by-step explanation:

The question involves finding the largest possible values of δ (delta) that correspond to ε (epsilon) values of 0.2 and 0.1, for the limit as x approaches 2 of the function f(x) = x3 − 4x + 9. According to the definition of the limit, we want to show that for each ε > 0, there exists a δ > 0 such that if 0 < |x - 2| < δ, then |f(x) - L| < ε, where L is the limit of f(x) as x approaches 2, which is given as 9.

To find the corresponding δ, we start by setting up the following inequality based on the given ε values:

|f(x) - L| = |x3 − 4x + 9 - 9| = |x3 − 4x| < ε.

Now we aim to find the δ that satisfies the inequality for ε = 0.2 and ε = 0.1. We can solve this inequality numerically or graphically. Upon solving, we will find the range for x that satisfies the ε condition, and this range will give us the value of δ such that 2-δ < x < 2+δ.

It is important to note that these calculations should be done carefully as achieving an exact numerical value for δ is not always straightforward and might require trial and error or graphical interpretation on a case-by-case basis.

Answer 2

Base on your question or in the illustration that you give if the the limit of the function is based on the illustration, what could be the possible largest value of delta if the epsilon is 0.2 and 0.1. So my answer is 0.02548 and 0.01238