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

(x) lim⁡x→+∞xx+1= \lim _{x \rightarrow+\infty} \frac{\sqrt{x}}{x+1}=

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Csc, sec, and cot of special anglesright chevron icon

Full solution

Q. (x) lim⁡x→+∞xx+1= \lim _{x \rightarrow+\infty} \frac{\sqrt{x}}{x+1}=
  1. Observation of Function Behavior: To find the limit of the function (x)/(x+1)(\sqrt{x})/(x+1) as xx approaches positive infinity, we can use the properties of limits and the behavior of functions as they approach infinity.\newlineWe can observe that as xx becomes very large, the term x\sqrt{x} grows slower than the linear term xx in the denominator. Therefore, we expect the fraction to approach zero.
  2. Dividing Numerator and Denominator: To formally evaluate the limit, we can divide the numerator and the denominator by xx to get the terms in a form that is easier to evaluate as xx approaches infinity.lim⁡x→+∞(xx+1)=lim⁡x→+∞(1/x1+(1/x))\lim_{x \rightarrow +\infty}\left(\frac{\sqrt{x}}{x+1}\right) = \lim_{x \rightarrow +\infty}\left(\frac{1/\sqrt{x}}{1+(1/x)}\right)
  3. Simplifying the Limit: Now, as xx approaches infinity, 1x\frac{1}{\sqrt{x}} approaches 00 and 1x\frac{1}{x} approaches 00. Therefore, the limit simplifies to: lim⁡x→+∞(1x)/(1+(1x))=01+0=0\lim_{x \to +\infty}\left(\frac{1}{\sqrt{x}}\right)/\left(1+\left(\frac{1}{x}\right)\right) = \frac{0}{1+0} = 0
  4. Final Answer: The final answer is that the limit of xx+1\frac{\sqrt{x}}{x+1} as xx approaches positive infinity is 00.

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