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The expression in the image appears to involve a limit calculation. Let me transcribe and analyze the problem step by step.

### Problem:
We are tasked to evaluate the limit:

\[
\lim_{x \to 0} \left( \frac{a^{x+1} - b^{x+1} - c^{x+1}}{a+b+c} \cdot \frac{1}{x} \right)
\]

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### Step 1: Approximation for small \( x \) using series expansion
For small values of \( x \), we can approximate the exponential terms \( a^{x+1} \), \( b^{x+1} \), and \( c^{x+1} \) using the formula:
\[
a^{x+1} = a \cdot a^x \approx a \cdot (1 + x \ln a), \quad \text{(since \( a^x \approx 1 + x \ln a \) for small \( x \))}
\]
Similarly:
\[
b^{x+1} \approx b \cdot (1 + x \ln b), \quad c^{x+1} \approx c \cdot (1 + x \ln c).
\]

---

### Step 2: Expand the numerator
Substitute the approximations into the numerator \( a^{x+1} - b^{x+1} - c^{x+1} \):
\[
a^{x+1} - b^{x+1} - c^{x+1} \approx a(1 + x \ln a) - b(1 + x \ln b) - c(1 + x \ln c).
\]
Simplify:
\[
a^{x+1} - b^{x+1} - c^{x+1} \approx a - b - c + x(a \ln a - b \ln b - c \ln c).
\]

---

### Step 3: Combine with the denominator
The denominator is \( a + b + c \). Thus, the full fraction becomes:
\[
\frac{a^{x+1} - b^{x+1} - c^{x+1}}{a + b + c} \approx \frac{(a - b - c) + x(a \ln a - b \ln b - c \ln c)}{a + b + c}.
\]

---

### Step 4: Multiply by \( \frac{1}{x} \)
When multiplying the expression by \( \frac{1}{x} \), the term \( \frac{a - b - c}{a + b + c} \) vanishes as \( x \to 0 \). The remaining term is:
\[
\frac{1}{x} \cdot \frac{x(a \ln a - b \ln b - c \ln c)}{a + b + c} = \frac{a \ln a - b \ln b - c \ln c}{a + b + c}.
\]

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### Final Answer:
\[
\lim_{x \to 0} \left( \frac{a^{x+1} - b^{x+1} - c^{x+1}}{a + b + c} \cdot \frac{1}{x} \right) = \frac{a \ln a - b \ln b - c \ln c}{a + b + c}.
\]

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Let me know if you would like me to clarify further or go over any step in detail!

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### Related Questions:
1. What is the Taylor series expansion for \( a^x \)?
2. How does the logarithmic approximation help in solving limits?
3. Can this method be generalized to similar types of limits?
4. What happens if \( a = b = c \) in the given problem?
5. How does the behavior of \( \ln(x) \) affect this type of calculation?

### Tip:
When solving limits involving exponential or logarithmic terms, approximations like \( a^x \approx 1 + x \ln a \) for small \( x \) can greatly simplify the problem.

Evaluate the limit: \( \lim_{x \to 0} \left( \frac{a^{x+1} - b^{x+1} - c^{x+1}}{a + b + c} \cdot \frac{1}{x} \right) \)

This article evaluates the limit \( \lim_{x \to 0} \left( \frac{a^{x+1} - b^{x+1} - c^{x+1}}{a + b + c} \cdot \frac{1}{x} \right) \) using series expansions and logarithmic approximations. Learn key concepts such as Taylor series and exponential function approximations.

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