For the limit below, find values of δ that correspond to the ε values.
lim_(x->0) (e^x - 1)/x = 1
𝜀	𝛿	
0.5	
 
Incorrect: Your answer is incorrect.
(Round your answer to three decimal places.)

We are given the limit:

\[
\lim_{{x \to 0}} \frac{e^x - 1}{x} = 1
\]

The goal is to find values of \( \delta \) that correspond to a given \( \epsilon \).

### Definition of Limit:
For the limit \( \lim_{{x \to 0}} f(x) = L \), we say that the limit equals \( L \) if, for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( 0 < |x - 0| < \delta \), the following holds:

\[
|f(x) - L| < \epsilon
\]

In this case, \( f(x) = \frac{e^x - 1}{x} \) and \( L = 1 \).

We need to satisfy:

\[
\left| \frac{e^x - 1}{x} - 1 \right| < \epsilon
\]

### Simplifying the Expression:
\[
\left| \frac{e^x - 1}{x} - 1 \right| = \left| \frac{e^x - 1 - x}{x} \right|
\]
So we need:

\[
\left| \frac{e^x - 1 - x}{x} \right| < \epsilon
\]

Now, we must find \( \delta \) in terms of \( \epsilon \).

### Step-by-Step Solution:

1. We need to solve for \( x \) such that \( \left| \frac{e^x - 1 - x}{x} \right| < \epsilon \).

2. Approximate \( e^x \) using the series expansion for small \( x \):

\[
e^x = 1 + x + \frac{x^2}{2} + O(x^3)
\]

Thus, \( e^x - 1 - x \approx \frac{x^2}{2} \), and we have:

\[
\left| \frac{\frac{x^2}{2}}{x} \right| = \left| \frac{x}{2} \right|
\]

3. We now need \( \left| \frac{x}{2} \right| < \epsilon \), which gives:

\[
|x| < 2\epsilon
\]

4. Therefore, we can choose \( \delta = 2\epsilon \).

### For \( \epsilon = 0.5 \):

\[
\delta = 2 \times 0.5 = 1.0
\]

Thus, the corresponding \( \delta \) for \( \epsilon = 0.5 \) is \( \delta = 1.0 \).

---

Would you like a more detailed explanation or have any other questions?

Here are 5 related questions to explore:

1. How do you find limits using the epsilon-delta definition?
2. What is the Taylor series expansion of \( e^x \)?
3. How do you solve for \( \delta \) in terms of \( \epsilon \) in general?
4. What are common limits involving exponential functions?
5. How does the approximation \( e^x \approx 1 + x \) for small \( x \) help in solving limits?

**Tip:** When using the epsilon-delta definition, approximating complex functions with Taylor series can greatly simplify the problem.

For the limit below, find values of δ that correspond to the ε values. lim_(x->0) (e^x - 1)/x = 1 ε δ 0.5 Incorrect: Your answer is incorrect. (Round your answer to three decimal places.)

Find δ corresponding to ε values for the limit lim_(x->0) (e^x - 1)/x = 1. This problem uses the epsilon-delta definition of limits, exponential function properties, and Taylor series expansion. Suitable for advanced high school and early college students.

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