Evaluate : `lim_(xrarroo)(a x^2+b x+c)/(dx^2+e x+f)dot`

Options

A. `(a)/(d)`
B. `(d)/(a)`
C. `(b)/(e)`
D. `(c)/(f)`

Explanation

To evaluate the limit

\[ \lim_{x \to \infty} \frac{ax^2 + bx + c}{dx^2 + ex + f}, \]

we will follow these steps:

### Step 1: Identify the highest power of \(x\)
In both the numerator and the denominator, the highest power of \(x\) is \(x^2\).

### Step 2: Factor out \(x^2\) from both the numerator and the denominator
We can factor \(x^2\) out of both the numerator and the denominator:

\[ \frac{ax^2 + bx + c}{dx^2 + ex + f} = \frac{x^2(a + \frac{b}{x} + \frac{c}{x^2})}{x^2(d + \frac{e}{x} + \frac{f}{x^2})}. \]

### Step 3: Simplify the expression
After factoring out \(x^2\), we can cancel \(x^2\) from the numerator and the denominator:

\[ = \frac{a + \frac{b}{x} + \frac{c}{x^2}}{d + \frac{e}{x} + \frac{f}{x^2}}. \]

### Step 4: Apply the limit as \(x\) approaches infinity
Now, we will apply the limit:

\[ \lim_{x \to \infty} \frac{a + \frac{b}{x} + \frac{c}{x^2}}{d + \frac{e}{x} + \frac{f}{x^2}}. \]

As \(x\) approaches infinity, the terms \(\frac{b}{x}\), \(\frac{c}{x^2}\), \(\frac{e}{x}\), and \(\frac{f}{x^2}\) will all approach \(0\). Thus, we have:

\[ = \frac{a + 0 + 0}{d + 0 + 0} = \frac{a}{d}. \]

### Final Answer
Therefore, the limit evaluates to:

\[ \lim_{x \to \infty} \frac{ax^2 + bx + c}{dx^2 + ex + f} = \frac{a}{d}. \]

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