Find the limit as x approaches negative infinity. lim_(x rarr-oo)(sqrt(9x^(2)+2))/(2x-9)=

Identify highest power of x: Identify the highest power of x in the numerator and denominator. In the expression (sqrt(9x^(2)+2))/(2x-9), the highest power of x in the numerator inside the square root is x^2, and in the denominator, it is x.Divide by highest power: Divide the numerator and the denominator by the highest power of x in the denominator. To simplify the limit, we divide both the numerator and the denominator by x. This gives us (sqrt(9 + 2/x^2))/(2 - 9/x).Take limit as x approaches: Take the limit as x approaches negative infinity. As x approaches negative infinity, the terms 2/x^2 and 9/x in the expression (sqrt(9 + 2/x^2))/(2 - 9/x) will approach 0. This simplifies the expression to (sqrt(9 + 0))/(2 - 0) which is sqrt(9)/2.Calculate simplified limit: Calculate the simplified limit. The simplified limit is sqrt(9)/2, which is 3/2.

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Algebra 1

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Q. Find the limit as

x

approaches negative infinity.

\newline

\lim _{x \rightarrow-\infty} \frac{\sqrt{9 x^{2}+2}}{2 x-9}=

Identify highest power of x: Identify the highest power of x in the numerator and denominator. $\newline$ In the expression $(\sqrt{9x^{2}+2})/(2x-9)$ , the highest power of x in the numerator inside the square root is $x^2$ , and in the denominator, it is $x$ .
Divide by highest power: Divide the numerator and the denominator by the highest power of $x$ in the denominator. $\newline$ To simplify the limit, we divide both the numerator and the denominator by $x$ . This gives us $\frac{\sqrt{9 + \frac{2}{x^2}}}{2 - \frac{9}{x}}$ .
Take limit as $x$ approaches: Take the limit as $x$ approaches negative infinity. $\newline$ As $x$ approaches negative infinity, the terms $\frac{2}{x^2}$ and $\frac{9}{x}$ in the expression $\frac{\sqrt{9 + \frac{2}{x^2}}}{2 - \frac{9}{x}}$ will approach $0$ . This simplifies the expression to $\frac{\sqrt{9 + 0}}{2 - 0}$ which is $\frac{\sqrt{9}}{2}$ .
Calculate simplified limit: Calculate the simplified limit. The simplified limit is $\sqrt{9}/2$ , which is $3/2$ .