To evaluate the limit [tex]\(\lim_ {x \to \infty} \sqrt{\frac{18 x^2 - 3 x + 2}{2 x^2 + 5}}\)[/tex], we can follow these steps:
1. Rewrite the expression inside the square root:
[tex]\[
\sqrt{\frac{18 x^2 - 3 x + 2}{2 x^2 + 5}}
\][/tex]
2. Factor out the highest power of [tex]\(x^2\)[/tex] from both the numerator and the denominator:
The highest power of [tex]\(x\)[/tex] in the numerator is [tex]\(x^2\)[/tex] and in the denominator is also [tex]\(x^2\)[/tex]. We factor [tex]\(x^2\)[/tex] out of both:
[tex]\[
= \sqrt{\frac{x^2(18 - \frac{3}{x} + \frac{2}{x^2})}{x^2(2 + \frac{5}{x^2})}}
\][/tex]
3. Simplify by canceling out the [tex]\(x^2\)[/tex] terms:
[tex]\[
= \sqrt{\frac{18 - \frac{3}{x} + \frac{2}{x^2}}{2 + \frac{5}{x^2}}}
\][/tex]
4. Analyze the limit as [tex]\( x \)[/tex] approaches infinity:
As [tex]\( x \to \infty \)[/tex], the terms with [tex]\( \frac{1}{x} \)[/tex] and [tex]\( \frac{1}{x^2} \)[/tex] approach zero. Therefore, we can simplify the fraction by ignoring these terms:
[tex]\[
\approx \sqrt{\frac{18 - 0 + 0}{2 + 0}} = \sqrt{\frac{18}{2}}
\][/tex]
5. Perform the final simplification:
[tex]\[
\sqrt{\frac{18}{2}} = \sqrt{9} = 3
\][/tex]
Therefore, the limit is:
[tex]\[
\lim_{x \rightarrow \infty} \sqrt{\frac{18 x^2 - 3 x + 2}{2 x^2 + 5}} = 3
\][/tex]
