(x) lim_(x rarr+oo)(sqrtx)/(x+1)=

Observation of Function Behavior: To find the limit of the function (sqrt(x))/(x+1) as x approaches positive infinity, we can use the properties of limits and the behavior of functions as they approach infinity. We can observe that as x becomes very large, the term sqrt(x) grows slower than the linear term x in the denominator. Therefore, we expect the fraction to approach zero.Dividing Numerator and Denominator: To formally evaluate the limit, we can divide the numerator and the denominator by x to get the terms in a form that is easier to evaluate as x approaches infinity. lim_(x rarr+oo)(sqrt(x))/(x+1) = lim_(x rarr+oo)(1/sqrt(x))/(1+(1/x))Simplifying the Limit: Now, as x approaches infinity, 1/sqrt(x) approaches 0 and 1/x approaches 0. Therefore, the limit simplifies to: lim_(x rarr+oo)(1/sqrt(x))/(1+(1/x)) = 0/(1+0) = 0Final Answer: The final answer is that the limit of (sqrt(x))/(x+1) as x approaches positive infinity is 0.

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(x) $\lim _{x \rightarrow+\infty} \frac{\sqrt{x}}{x+1}=$

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\lim _{x \rightarrow+\infty} \frac{\sqrt{x}}{x+1}=

Observation of Function Behavior: To find the limit of the function $(\sqrt{x})/(x+1)$ as $x$ approaches positive infinity, we can use the properties of limits and the behavior of functions as they approach infinity. $\newline$ We can observe that as $x$ becomes very large, the term $\sqrt{x}$ grows slower than the linear term $x$ in the denominator. Therefore, we expect the fraction to approach zero.
Dividing Numerator and Denominator: To formally evaluate the limit, we can divide the numerator and the denominator by $x$ to get the terms in a form that is easier to evaluate as $x$ approaches infinity. $\lim_{x \rightarrow +\infty}\left(\frac{\sqrt{x}}{x+1}\right) = \lim_{x \rightarrow +\infty}\left(\frac{1/\sqrt{x}}{1+(1/x)}\right)$
Simplifying the Limit: Now, as $x$ approaches infinity, $\frac{1}{\sqrt{x}}$ approaches $0$ and $\frac{1}{x}$ approaches $0$ . Therefore, the limit simplifies to: $\lim_{x \to +\infty}\left(\frac{1}{\sqrt{x}}\right)/\left(1+\left(\frac{1}{x}\right)\right) = \frac{0}{1+0} = 0$
Final Answer: The final answer is that the limit of $\frac{\sqrt{x}}{x+1}$ as $x$ approaches positive infinity is $0$ .