Question

find the maximum value of \\(\delta > 0\\) that satisfies the limit claim corresponding to \\(\varepsilon = 0.4\\), that is, such that \\(0 < |x - c| < \delta\\) would imply \\(|f(x) - l| < 0.4\\).

\\\lim_{x \to 2} (4x - 3) = 5\\

Explanation:

Identify the given limit and parameters

We are given the limit claim:

$$ \lim_{x \to 2} (4x - 3) = 5 $$

Here, the function is \(f(x) = 4x - 3\), the limit point is \(c = 2\), and the limit value is \(L = 5\). We are given \(\varepsilon = 0.4\).

Set up the inequality

We want to find the maximum value of \(\delta > 0\) such that \(0 < |x - c| < \delta\) implies \(|f(x) - L| < \varepsilon\).
Using Epsilon-Delta Definition, we substitute our values into the inequality:

$$ |(4x - 3) - 5| < 0.4 $$

Simplify the function inequality

Simplify the expression inside the absolute value:

$$ |4x - 8| < 0.4 $$

Factor out the constant \(4\):

$$ 4|x - 2| < 0.4 $$

Solve for the delta inequality

Divide both sides of the inequality by \(4\):

$$ |x - 2| < \frac{0.4}{4} $$

$$ |x - 2| < 0.1 $$

Determine the maximum delta

Comparing \(|x - 2| < 0.1\) with the definition \(0 < |x - 2| < \delta\), we find that any \(\delta \le 0.1\) satisfies the condition.
Therefore, the maximum value of \(\delta\) is \(0.1\).

Answer:

Find the maximum value of \(\delta > 0\) that satisfies the limit claim corresponding to \(\varepsilon = 0.4\), that is, such that \(0 < |x - c| < \delta\) would imply \(|f(x) - L| < 0.4\).

$$ \lim_{x \to 2} (4x - 3) = 5 $$

Maximum value of \(\delta\): <blank>0.1</blank>