Question

evaluate the limit using the appropriate properties of limits. (if the limit is infini (lim_{x \to infty} \frac{8x^2 - 5}{3x^2 + x - 3})

Explanation:

Step1: Divide numerator and denominator by \(x^2\)

To evaluate the limit as \(x\) approaches infinity, we divide each term in the numerator and the denominator by the highest power of \(x\) in the denominator, which is \(x^2\).
\[
\lim_{x \to \infty} \frac{8x^2 - 5}{3x^2 + x - 3} = \lim_{x \to \infty} \frac{\frac{8x^2}{x^2} - \frac{5}{x^2}}{\frac{3x^2}{x^2} + \frac{x}{x^2} - \frac{3}{x^2}}
\]

Step2: Simplify each term

Simplify each fraction:
\[
\lim_{x \to \infty} \frac{8 - \frac{5}{x^2}}{3 + \frac{1}{x} - \frac{3}{x^2}}
\]

Step3: Evaluate the limit of each term

As \(x \to \infty\), \(\frac{1}{x} \to 0\) and \(\frac{1}{x^2} \to 0\). So we substitute these limits into the expression:
\[
\frac{\lim_{x \to \infty} 8 - \lim_{x \to \infty} \frac{5}{x^2}}{\lim_{x \to \infty} 3 + \lim_{x \to \infty} \frac{1}{x} - \lim_{x \to \infty} \frac{3}{x^2}} = \frac{8 - 0}{3 + 0 - 0}
\]

Step4: Simplify the fraction

Simplify the numerator and the denominator:
\[
\frac{8}{3}
\]

Answer:

\(\frac{8}{3}\)