Question

evaluate the limit \\(\lim_{x \to \infty} \frac{3 + 3x}{2 - 2x}\\)

Explanation:

Step1: Divide numerator and denominator by \( x \)

To evaluate the limit as \( x \to \infty \), we divide each term in the numerator and the denominator by the highest power of \( x \) in the denominator, which is \( x \). So we have:

$$ \lim_{x \to \infty} \frac{\frac{3}{x} + \frac{3x}{x}}{\frac{2}{x} - \frac{2x}{x}} $$

Step2: Simplify each term

Simplify each fraction:

$$ \lim_{x \to \infty} \frac{\frac{3}{x} + 3}{\frac{2}{x} - 2} $$

Step3: Evaluate the limit as \( x \to \infty \)

As \( x \to \infty \), \( \frac{3}{x} \to 0 \) and \( \frac{2}{x} \to 0 \) because the reciprocal of a very large number approaches 0. Substituting these limits in, we get:

$$ \frac{0 + 3}{0 - 2} = \frac{3}{-2}=-\frac{3}{2} $$

Answer:

\( -\frac{3}{2} \)