To solve the limit \(\lim_{x \to 0} \frac{(e^x - 1) \tan{x}}{x^2}\), we will use a detailed step-by-step approach.
### Step 1: Understand the Components
We need to find the limit of the function \(\frac{(e^x - 1) \tan{x}}{x^2}\) as \(x\) approaches 0.
### Step 2: Break Down the Expression
Firstly, let's analyze the individual components of the expression:
- \(e^x - 1\)
- \(\tan{x}\)
- \(x^2\)
### Step 3: Use Series Expansions
For small values of \(x\), we can use the series expansions of these functions to approximate their behavior near 0.
- The Taylor series expansion for \(e^x\) around \(x = 0\) is:
[tex]\[
e^x \approx 1 + x + \frac{x^2}{2!} + O(x^3)
\][/tex]
Hence,
[tex]\[
e^x - 1 \approx x + \frac{x^2}{2} + O(x^3)
\][/tex]
- The Taylor series expansion for \(\tan{x}\) around \(x = 0\) is:
[tex]\[
\tan{x} \approx x + \frac{x^3}{3} + O(x^5)
\][/tex]
### Step 4: Substitute the Series Expansions
Now, substitute these series expansions into the original limit expression:
[tex]\[
\frac{(e^x - 1) \tan{x}}{x^2} \approx \frac{\left(x + \frac{x^2}{2} + O(x^3)\right) \left(x + \frac{x^3}{3} + O(x^5)\right)}{x^2}
\][/tex]
### Step 5: Simplify the Expression
Let's multiply the series expansions in the numerator:
[tex]\[
\left(x + \frac{x^2}{2}\right) \left(x + \frac{x^3}{3}\right) \approx x^2 + \frac{x^4}{3} + \frac{x^2 \cdot x}{2} + O(x^5)
\][/tex]
[tex]\[
= x^2 + \frac{x^3}{2} + \frac{x^4}{3} + O(x^5)
\][/tex]
Now, divide each term in the numerator by \(x^2\):
[tex]\[
\frac{x^2 + \frac{x^3}{2} + \frac{x^4}{3} + O(x^5)}{x^2} = 1 + \frac{x}{2} + \frac{x^2}{3} + O(x^3)
\][/tex]
### Step 6: Evaluate the Limit
As \(x \to 0\), all the higher-order terms \(O(x)\), \(O(x^2)\), etc., will approach 0. Hence, the dominant term is just 1.
Therefore,
[tex]\[
\lim_{x \to 0} \frac{(e^x - 1) \tan{x}}{x^2} = 1
\][/tex]
So the limit is [tex]\(\boxed{1}\)[/tex].
