How do you find a delta for a given epsilon using a concrete numerical example?

Conditions

• A specific function graph is available.
• Values for $L$, $a$, and $\epsilon$ are given.
• The function is continuous or has a limit at $a$.

Reasoning, step by step

1. Set the limit $L$ and approach point $a$ (e.g., $L=2, a=1$).
2. Choose an epsilon (e.g., $\epsilon=0.5$).
3. Calculate the target y-range: $(2 - 0.5, 2 + 0.5) = (1.5, 2.5)$.
4. Inspect the graph to find the x-range that produces y-values in $(1.5, 2.5)$. Suppose this is $(0.9, 1.1)$.
5. Calculate the distance from the center $a=1$ to the endpoints: $|1 - 0.9| = 0.1$ and $|1.1 - 1| = 0.1$.
6. Select the smaller distance as delta: $\delta = 0.1$.
7. Verify that if $0 < |x - 1| < 0.1$, then $|f(x) - 2| < 0.5$.

Example

In the video, for $L=2, a=1, \epsilon=0.5$, the speaker identifies the x-interval $(0.9, 1.1)$ on the graph. The distance from 1 to 0.9 (or 1.1) is 0.1, so $\delta=0.1$ is a valid choice.

Common misconceptions

• Thinking that delta must be unique; any smaller positive delta also works.
• Assuming the method works without looking at the specific function graph; delta depends on the function's behavior.
• Believing that finding one delta proves the limit; it must work for *every* epsilon.

Watch the explanation

Explore next

• Why does the definition require this to work for every epsilon, not just one?
• How would the delta change if epsilon were smaller?
• What is the algebraic relationship between epsilon and delta for a linear function?