Evaluate : ` underset( x to prop) lim (2x^(2) + 7x+ 5)/( 4x^(2) +3x- 1 ) `

Answer

To evaluate the limit as \( x \) approaches infinity for the expression \( \frac{2x^2 + 7x + 5}{4x^2 + 3x - 1} \), we can 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 \( x^2 \)
We can factor \( x^2 \) out of both the numerator and the denominator:

\[ \frac{2x^2 + 7x + 5}{4x^2 + 3x - 1} = \frac{x^2(2 + \frac{7}{x} + \frac{5}{x^2})}{x^2(4 + \frac{3}{x} - \frac{1}{x^2})} \]

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

\[ = \frac{2 + \frac{7}{x} + \frac{5}{x^2}}{4 + \frac{3}{x} - \frac{1}{x^2}} \]

### Step 4: Take the limit as \( x \) approaches infinity
As \( x \) approaches infinity, the terms \( \frac{7}{x} \), \( \frac{5}{x^2} \), \( \frac{3}{x} \), and \( -\frac{1}{x^2} \) all approach 0. Therefore, we can substitute these limits into the expression:

\[ = \frac{2 + 0 + 0}{4 + 0 - 0} = \frac{2}{4} \]

### Step 5: Simplify the result
Now, we simplify \( \frac{2}{4} \):

\[ = \frac{1}{2} \]

### Final Answer
Thus, the limit is:

\[ \lim_{x \to \infty} \frac{2x^2 + 7x + 5}{4x^2 + 3x - 1} = \frac{1}{2} \]

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