Find the minimum value of

k

such that

δ=

ε/k

satisfies the

ε-δ

definition of the following claim. Write the exact answer. Do not round.

lim

x→5

((4-(x/5))=3

We are asked to find the minimum value of \( k \) such that \( \delta = \frac{\varepsilon}{k} \) satisfies the \( \varepsilon \)-\( \delta \) definition of the given limit:

\[
\lim_{x \to 5} \left( 4 - \frac{x}{5} \right) = 3.
\]

### Step 1: Express the limit in standard form

We want to show that:

\[
\lim_{x \to 5} f(x) = 3
\]
where \( f(x) = 4 - \frac{x}{5} \).

This means for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - 5| < \delta \), then \( |f(x) - 3| < \varepsilon \).

### Step 2: Work with the function \( f(x) \)

We need to manipulate \( |f(x) - 3| \):

\[
|f(x) - 3| = \left| \left( 4 - \frac{x}{5} \right) - 3 \right| = \left| 1 - \frac{x}{5} \right|.
\]

### Step 3: Simplify the inequality

Now we need to ensure that \( |1 - \frac{x}{5}| < \varepsilon \). To simplify this inequality:

\[
\left| 1 - \frac{x}{5} \right| = \left| \frac{5 - x}{5} \right| = \frac{|x - 5|}{5}.
\]

So, the condition becomes:

\[
\frac{|x - 5|}{5} < \varepsilon.
\]

### Step 4: Solve for \( |x - 5| \)

Multiply both sides of the inequality by 5:

\[
|x - 5| < 5\varepsilon.
\]

### Step 5: Set \( \delta \)

We see that to satisfy the \( \varepsilon \)-\( \delta \) definition, we must have \( \delta = 5\varepsilon \). Therefore, we want:

\[
\delta = \frac{\varepsilon}{k}.
\]

Equating this to our expression for \( \delta \), we get:

\[
\frac{\varepsilon}{k} = 5\varepsilon.
\]

### Step 6: Solve for \( k \)

Dividing both sides by \( \varepsilon \) (assuming \( \varepsilon > 0 \)):

\[
\frac{1}{k} = 5 \quad \Rightarrow \quad k = \frac{1}{5}.
\]

### Final Answer:

The minimum value of \( k \) is \( \boxed{\frac{1}{5}} \).

---

Do you want any further details or have any questions?

Here are some related questions to explore:

1. What is the general form of the \( \varepsilon \)-\( \delta \) definition of a limit?
2. How does the choice of \( \delta \) depend on the function \( f(x) \)?
3. Can you provide an example of a limit where \( \delta \) depends non-linearly on \( \varepsilon \)?
4. What happens if the limit involves infinity instead of a finite value?
5. How does the \( \varepsilon \)-\( \delta \) definition extend to multivariable limits?

**Tip:** When using the \( \varepsilon \)-\( \delta \) definition, always carefully manipulate the inequalities involving \( \varepsilon \) to find the correct \( \delta \).

Find the minimum value of k such that δ = ε/k satisfies the ε-δ definition of the following claim. Write the exact answer. Do not round.
lim x→5 ((4-(x/5))=3

Explore how to find the minimum value of k such that δ = ε/k satisfies the epsilon-delta definition for the limit of (4 - x/5) as x approaches 5. The solution involves applying the epsilon-delta theorem and algebraic manipulation. Perfect for students studying calculus or real analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation