7. lim _(xarrow infty )(5x^3+3x^2+2)/(3x^2)+9 -

To find the limit of the given expression as \( x \) approaches infinity, we can use the technique of dividing both the numerator and denominator by the highest power of \( x \) in the denominator.<br /><br />Given expression:<br />\[ \lim_{x \to \infty} \frac{5x^3 + 3x^2 + 2}{3x^2 + 9} \]<br /><br />Step 1: Divide both the numerator and the denominator by \( x^2 \), the highest power of \( x \) in the denominator.<br /><br />\[ \lim_{x \to \infty} \frac{\frac{5x^3 + 3x^2 + 2}{x^2}}{\frac{3x^2 + 9}{x^2}} \]<br /><br />Step 2: Simplify the expression.<br /><br />\[ \lim_{x \to \infty} \frac{5x + 3 + \frac{2}{x^2}}{3 + \frac{9}{x^2}} \]<br /><br />Step 3: As \( x \) approaches infinity, the terms \( \frac{2}{x^2} \) and \( \frac{9}{x^2} \) approach 0.<br /><br />\[ \lim_{x \to \infty} \frac{5x + 3 + 0}{3 + 0} \]<br /><br />Step 4: Simplify the limit expression.<br /><br />\[ \lim_{x \to \infty} \frac{5x + 3}{3} \]<br /><br />Step 5: As \( x \) approaches infinity, \( 5x + 3 \) also approaches infinity.<br /><br />\[ \lim_{x \to \infty} \frac{5x + 3}{3} = \infty \]<br /><br />Therefore, the limit of the given expression as \( x \) approaches infinity is \( \infty \).