https://file.aiquickdraw.com/mathbot/file/1727420213149m7agt9wv.jpg

The expression in the image represents a limit problem:

\[
\lim_{{x \to 1}} \frac{x^n - (n+1)x + n}{x^m - (m+1)x + m}
\]

To solve this limit, let's analyze it step by step:

### 1. **Substitute \(x = 1\) directly**:
   Substituting \(x = 1\) in both the numerator and the denominator:
   - Numerator: \(1^n - (n+1) \cdot 1 + n = 1 - n - 1 + n = 0\)
   - Denominator: \(1^m - (m+1) \cdot 1 + m = 1 - m - 1 + m = 0\)

So, both the numerator and denominator result in 0, giving us the indeterminate form \(\frac{0}{0}\). This suggests using **L'Hopital's Rule**, which applies to limits of the form \(\frac{0}{0}\).

### 2. **Apply L'Hopital's Rule**:
   Differentiate the numerator and the denominator with respect to \(x\):
   - **Numerator**:
     \[
     \frac{d}{dx} \left( x^n - (n+1)x + n \right) = n x^{n-1} - (n+1)
     \]
   - **Denominator**:
     \[
     \frac{d}{dx} \left( x^m - (m+1)x + m \right) = m x^{m-1} - (m+1)
     \]

### 3. **Substitute \(x = 1\) again**:
   After applying L'Hopital's Rule, substitute \(x = 1\) into the derivatives:
   - Numerator: \(n \cdot 1^{n-1} - (n+1) = n - (n+1) = -1\)
   - Denominator: \(m \cdot 1^{m-1} - (m+1) = m - (m+1) = -1\)

Thus, the limit becomes:
\[
\lim_{{x \to 1}} \frac{-1}{-1} = 1
\]

### Final Answer:
\[
\boxed{1}
\]

Would you like more details or have any questions?

Here are five related questions:
1. How does L'Hopital's Rule help when dealing with indeterminate forms?
2. What other techniques can be used to solve limits when direct substitution fails?
3. Why is the derivative of \(x^n\) equal to \(n x^{n-1}\)?
4. What is the significance of indeterminate forms like \(\frac{0}{0}\) in calculus?
5. How can limits be solved without using L'Hopital's Rule?

**Tip:** L'Hopital's Rule can be applied to other indeterminate forms, such as \(\frac{\infty}{\infty}\), making it a versatile tool in solving limits.

Find the limit of the expression (x^n - (n+1)x + n) / (x^m - (m+1)x + m) as x approaches 1.

Learn how to solve the limit of the expression (x^n - (n+1)x + n) / (x^m - (m+1)x + m) as x approaches 1 using L'Hopital's Rule. This problem involves finding derivatives and analyzing indeterminate forms (0/0). Suitable for students in undergraduate calculus.

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