Find the limit as x approaches positive infinity. lim_(x rarr oo)(2x^(4)-7)/(sqrt(4x^(8)+7x^(5)))=

Identify highest power of x: Identify the highest power of x in the numerator and the denominator. In the numerator, the highest power of x is x^4. In the denominator, after taking the square root, the highest power of x is x^4 (since sqrt(x^8) = x^4).Divide numerator and denominator: Divide the numerator and the denominator by x^4, the highest power of x found in the previous step. lim_(x rarr oo)(2x^(4)/x^(4) - 7/x^(4))/(sqrt(4x^(8)/x^(8) + 7x^(5)/x^(8)))Simplify expression: Simplify the expression by canceling out the x terms where possible. lim_(x rarr oo)(2 - 7/x^(4))/(sqrt(4 + 7/x^(3)))Approach positive infinity: As x approaches positive infinity, the terms with x in the denominator approach zero. lim_(x rarr oo)(2 - 0)/(sqrt(4 + 0)) = 2/sqrt(4)Calculate simplified expression: Calculate the simplified expression. 2/sqrt(4) = 2/2 = 1

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Algebra 1

Powers with decimal and fractional bases

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Q. Find the limit as

x

approaches positive infinity.

\newline

\lim _{x \rightarrow \infty} \frac{2 x^{4}-7}{\sqrt{4 x^{8}+7 x^{5}}}=

Identify highest power of $x$ : Identify the highest power of $x$ in the numerator and the denominator. $\newline$ In the numerator, the highest power of $x$ is $x^4$ . In the denominator, after taking the square root, the highest power of $x$ is $x^4$ (since $\sqrt{x^8} = x^4$ ).
Divide numerator and denominator: Divide the numerator and the denominator by $x^4$ , the highest power of $x$ found in the previous step. $\newline$ $\lim_{x \rightarrow \infty}\left(\frac{2x^{4}}{x^{4}} - \frac{7}{x^{4}}\right)/\left(\sqrt{\frac{4x^{8}}{x^{8}} + \frac{7x^{5}}{x^{8}}}\right)$
Simplify expression: Simplify the expression by canceling out the $x$ terms where possible. $\lim_{x \to \infty}\frac{2 - \frac{7}{x^{4}}}{\sqrt{4 + \frac{7}{x^{3}}}}$
Approach positive infinity: As $x$ approaches positive infinity, the terms with $x$ in the denominator approach zero. $\lim_{x \rightarrow \infty}\frac{2 - 0}{\sqrt{4 + 0}} = \frac{2}{\sqrt{4}}$
Calculate simplified expression: Calculate the simplified expression. $\frac{2}{\sqrt{4}} = \frac{2}{2} = 1$