`lim_(xrarr1) (ax^(2)+bx+c)/(cx^(2)+bx+a)a+b+cne0`

Explanation

To solve the limit problem \( \lim_{x \to 1} \frac{ax^2 + bx + c}{cx^2 + bx + a} \) given that \( a + b + c \neq 0 \), we will follow these steps:

### Step 1: Substitute \( x = 1 \) into the function
We start by substituting \( x = 1 \) into the expression:

\[ \frac{a(1)^2 + b(1) + c}{c(1)^2 + b(1) + a} \]

### Step 2: Simplify the expression
This simplifies to:

\[ \frac{a + b + c}{c + b + a} \]

### Step 3: Evaluate the limit
Since both the numerator and the denominator are equal, we have:

\[ \frac{a + b + c}{a + b + c} = 1 \]

### Step 4: Conclusion
Thus, the limit is:

\[ \lim_{x \to 1} \frac{ax^2 + bx + c}{cx^2 + bx + a} = 1 \]

### Final Answer
The limit is \( 1 \).

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