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[tex]\lim_{ x \to \infty} \frac{ \sqrt{x^3+7x^2} -\sqrt{x^3}} {\sqrt{x}}[/tex]
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$$\begin{align*} \lim_{ x \to \infty} \frac{ \sqrt{x^3+7x^2} -\sqrt{x^3}} {\sqrt{x}} & = \lim_{x\to\infty}\left(\sqrt{x^2 + 7x} - \sqrt{x^2}\right) \\
& = \lim_{x\to\infty}\left(\sqrt{x^2 + 7x} - \sqrt{x^2}\right)\frac{\sqrt{x^2 + 7x} +\sqrt{x^2}}{\sqrt{x^2 + 7x} + \sqrt{x^2}} \\
& = \lim_{x\to\infty}\frac{(x^2 + 7x) - x^2}{\sqrt{x^2 + 7x} + \sqrt{x^2}} \\
& = \lim_{x\to\infty}\frac{7x}{\sqrt{x^2 + 7x} + x} \\
& = \lim_{x\to\infty}\frac{\frac{1}{x}7x}{\frac{1}{x}\left(\sqrt{x^2 + 7x} + x\right)} \\
& = \lim_{x\to\infty}\frac{7}{\sqrt{1 + \frac{7}{x}} + 1} \\
& = \frac72.\end{align*}$$
& = \lim_{x\to\infty}\left(\sqrt{x^2 + 7x} - \sqrt{x^2}\right)\frac{\sqrt{x^2 + 7x} +\sqrt{x^2}}{\sqrt{x^2 + 7x} + \sqrt{x^2}} \\
& = \lim_{x\to\infty}\frac{(x^2 + 7x) - x^2}{\sqrt{x^2 + 7x} + \sqrt{x^2}} \\
& = \lim_{x\to\infty}\frac{7x}{\sqrt{x^2 + 7x} + x} \\
& = \lim_{x\to\infty}\frac{\frac{1}{x}7x}{\frac{1}{x}\left(\sqrt{x^2 + 7x} + x\right)} \\
& = \lim_{x\to\infty}\frac{7}{\sqrt{1 + \frac{7}{x}} + 1} \\
& = \frac72.\end{align*}$$