`lim_(x to infty) (x^(3)-3x+2)/(2x^(3)+x-3)` =

Options

A. 2
B. `1//2`
C. 0
D. not exist

Answer

To solve the limit \( \lim_{x \to \infty} \frac{x^3 - 3x + 2}{2x^3 + x - 3} \), 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 \( x^3 \).

### Step 2: Factor out the highest power of \( x \)
We can factor \( x^3 \) out of both the numerator and the denominator:

\[ \lim_{x \to \infty} \frac{x^3(1 - \frac{3}{x^2} + \frac{2}{x^3})}{x^3(2 + \frac{1}{x^2} - \frac{3}{x^3})} \]

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

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

### Step 4: Evaluate the limit
Now, we can evaluate the limit as \( x \) approaches infinity. As \( x \to \infty \), the terms \( \frac{3}{x^2} \), \( \frac{2}{x^3} \), \( \frac{1}{x^2} \), and \( \frac{3}{x^3} \) all approach 0:

\[ \lim_{x \to \infty} \frac{1 - 0 + 0}{2 + 0 - 0} = \frac{1}{2} \]

### Conclusion
Thus, the limit is:

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