To estimate the limit [tex]\(\lim_{x \to 0} \frac{e^x - 1}{x}\)[/tex], we can create a table of values for the function [tex]\(f(x) = \frac{e^x - 1}{x}\)[/tex] at specific [tex]\(x\)[/tex] values that approach 0 from both the negative and positive sides.
Here are the [tex]\(x\)[/tex] values we will use and their corresponding [tex]\(f(x)\)[/tex] values:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & 0.1 \\
\hline
f(x) & 0.9516 & 0.9950 & 0.9995 & 1.0005 & 1.0050 & 1.0517 \\
\hline
\end{array}
\][/tex]
### Step-by-Step Solution
1. Evaluate [tex]\(f(x)\)[/tex] at given [tex]\(x\)[/tex] values:
For [tex]\(x = -0.1\)[/tex]:
[tex]\[
f(-0.1) = 0.9516
\][/tex]
For [tex]\(x = -0.01\)[/tex]:
[tex]\[
f(-0.01) = 0.9950
\][/tex]
For [tex]\(x = -0.001\)[/tex]:
[tex]\[
f(-0.001) = 0.9995
\][/tex]
For [tex]\(x = 0.001\)[/tex]:
[tex]\[
f(0.001) = 1.0005
\][/tex]
For [tex]\(x = 0.01\)[/tex]:
[tex]\[
f(0.01) = 1.0050
\][/tex]
For [tex]\(x = 0.1\)[/tex]:
[tex]\[
f(0.1) = 1.0517
\][/tex]
2. Observe behavior of [tex]\(f(x)\)[/tex] as [tex]\(x\)[/tex] approaches 0:
When looking at the values, we see:
[tex]\[
\begin{array}{cc}
x & f(x) \\
-0.1 & 0.9516 \\
-0.01 & 0.9950 \\
-0.001 & 0.9995 \\
0.001 & 1.0005 \\
0.01 & 1.0050 \\
0.1 & 1.0517 \\
\end{array}
\][/tex]
As [tex]\(x\)[/tex] gets closer to 0 (from either side), the function values [tex]\(f(x)\)[/tex] are getting closer to 1.
3. Estimate the limit:
Based on the values in the table, we observe that as [tex]\(x\)[/tex] approaches 0 from both the negative and positive sides, [tex]\(f(x)\)[/tex] approaches 1. Thus, the estimated limit is:
[tex]\[
\lim_{x \to 0} \frac{e^x - 1}{x} \approx 1
\][/tex]
### Conclusion
From the table of values, we can reasonably conclude that the limit as [tex]\(x\)[/tex] approaches 0 for the function [tex]\(\frac{e^x - 1}{x}\)[/tex] is 1. This method of using a table of values gives us a good approximation and visual evidence of the limit behavior.
