Question

write the appropriate \\(\epsilon\\)-\\(\delta\\) definition for the given statement.

\\\lim_{x \to a} f(x) = l\\

enter \e\ for \\(\epsilon\\) and \d\ for \\(\delta\\).

for every \\(> 0\\), there exists a \\(> 0\\),
such that if \\(0 < |\\) \\(| <\\) ,
then \\(|f(x) -\\) \\(| <\\) .

Explanation:

Identify the standard definition

The standard formal definition of a limit, \(\lim_{x\to a} f(x) = L\), states:
For every \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x - a| < \delta\), then \(|f(x) - L| < \epsilon\).

Map to the input fields

The problem asks to write the appropriate \(\epsilon\)-\(\delta\) definition using "e" for \(\epsilon\) and "d" for \(\delta\).
Let's map each blank in the given template:

• "For every [Blank 1] > 0": This corresponds to \(\epsilon\), so we enter "e".
• "there exists a [Blank 2] > 0": This corresponds to \(\delta\), so we enter "d".
• "such that if \(0 < | [Blank 3] | < [Blank 4]\)": This corresponds to \(0 < |x - a| < \delta\), so [Blank 3] is "x - a" (or "x-a") and [Blank 4] is "d".
• "then \(| f(x) - [Blank 5] | < [Blank 6]\)": This corresponds to \(|f(x) - L| < \epsilon\), so [Blank 5] is "L" and [Blank 6] is "e".

Verify the expressions

Let's double check the exact template from the image:
"For every [ ] > 0, there exists a [ ] > 0,
such that if 0 < | [ ] | < [ ],
then | f(x) - [ ] | < [ ]."

Plugging in the mapped values:

• First blank: e
• Second blank: d
• Third blank: x-a
• Fourth blank: d
• Fifth blank: L
• Sixth blank: e

Answer:

Write the appropriate \(\epsilon\)-\(\delta\) definition for the given statement.

\(\lim_{x\to a} f(x) = L\)

Enter "e" for \(\epsilon\) and "d" for \(\delta\).

For every <blank>e</blank> \(> 0\), there exists a <blank>d</blank> \(> 0\),
such that if \(0 < | \)<blank>\(x-a\)</blank>\( | < \)<blank>d</blank>,
then \(| f(x) - \)<blank>L</blank>\( | < \)<blank>e</blank>.