(*) lim_(x rarr oo)(sqrtx)/(x)

Understand the problem: Understand the limit problem. We need to find the limit of the function (sqrt(x))/(x) as x approaches infinity. This is a case of an indeterminate form since both the numerator and the denominator are growing without bounds.Simplify the expression: Simplify the expression. To simplify the expression, we can divide both the numerator and the denominator by x. However, since x is under a square root in the numerator, we divide by sqrt(x) instead to keep the expression equivalent. (sqrt(x))/(x) = (sqrt(x)/sqrt(x))/(x/sqrt(x)) = 1/(sqrt(x))Evaluate the limit: Evaluate the limit. Now we need to evaluate the limit of 1/(sqrt(x)) as x approaches infinity. As x becomes larger and larger, the denominator sqrt(x) will also become larger, making the whole fraction smaller and smaller. lim_(x -> oo) 1/(sqrt(x)) = 0Conclude the solution: Conclude the solution. Since the limit of 1/(sqrt(x)) as x approaches infinity is 0, the original limit of (sqrt(x))/(x) as x approaches infinity is also 0.

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(*) $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt{x}}{x}$

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Algebra 2

Simplify radical expressions involving fractions

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\newline

\lim _{x \rightarrow \infty} \frac{\sqrt{x}}{x}

Understand the problem: Understand the limit problem. $\newline$ We need to find the limit of the function $(\sqrt{x})/(x)$ as $x$ approaches infinity. This is a case of an indeterminate form since both the numerator and the denominator are growing without bounds.
Simplify the expression: Simplify the expression. $\newline$ To simplify the expression, we can divide both the numerator and the denominator by $x$ . However, since $x$ is under a square root in the numerator, we divide by $\sqrt{x}$ instead to keep the expression equivalent. $\newline$ $\frac{\sqrt{x}}{x} = \frac{\sqrt{x}/\sqrt{x}}{x/\sqrt{x}} = \frac{1}{\sqrt{x}}$
Evaluate the limit: Evaluate the limit. $\newline$ Now we need to evaluate the limit of $\frac{1}{\sqrt{x}}$ as $x$ approaches infinity. As $x$ becomes larger and larger, the denominator $\sqrt{x}$ will also become larger, making the whole fraction smaller and smaller. $\newline$ $\lim_{x \to \infty} \frac{1}{\sqrt{x}} = 0$
Conclude the solution: Conclude the solution. $\newline$ Since the limit of $\frac{1}{\sqrt{x}}$ as $x$ approaches infinity is $0$ , the original limit of $\frac{\sqrt{x}}{x}$ as $x$ approaches infinity is also $0$ .