babipsylon 0 Posted July 28, 2016 Share Posted July 28, 2016 I need this limit ((x(sqrt(x+2)))/sqrt(x+1))-x to calculate the asymptote of this function: ((x(sqrt(x+2)))/sqrt(x+1)). which, according to the class notes: y=x+1/2 with a = 1, b = 1/2 However, the online math calculators say that currently no steps are supported to show for this kind of problem. I calculated this limit as x*sqrt(1)-x = 1, but apparently the correct answer is 1/2. What is my mistake? Thx in advance. r Link to post Share on other sites

mathematic 104 Posted July 28, 2016 Share Posted July 28, 2016 [latex]\frac{x(\sqrt{x+2})}{\sqrt{x+1}}-x=\frac{x}{\sqrt{x+1}}(\sqrt{x+2}-\sqrt{x+1})= \frac{x}{\sqrt{x+1}(\sqrt{x+2}+\sqrt{x+1})}(x+2-(x+1))[/latex] Limit = 1/2, since the denominator ->2x. 1 Link to post Share on other sites

renerpho 9 Posted September 28, 2016 Share Posted September 28, 2016 (edited) You should use l'Hôpital's rule for a rigorous proof:[math]\lim_{x\to\infty} {\frac{x\sqrt{x+2}}{\sqrt{x+1}}-x}=\lim_{x\to\infty} {\frac{x}{\sqrt{(x+1)(x+2)}+x+1}}[/math] (see the previous post). Using l'Hôpital's rule, this equals [math]\lim_{x\to\infty} {\frac{1}{(\sqrt{(x+1)(x+2)})'+1}}=\lim_{x\to\infty} {\frac{1}{\frac{2x+3}{2\sqrt{(x+1)(x+2)}}+1}}[/math]. Assuming the limit exists, this is equal to [math]\frac{1}{\lim_{x\to\infty} {\frac{2x+3}{2\sqrt{(x+1)(x+2)}}}+1}[/math]. You then evaluate [math]\lim_{x\to\infty} {\frac{2x+3}{2\sqrt{(x+1)(x+2)}}}[/math]: Using l'Hôpital again, this is equal to [math]\lim_{x\to\infty} {\frac{2}{\frac{2(2x+3)}{2\sqrt{(x+1)(x+2)}}}}=\lim_{x\to\infty} {\frac{2\sqrt{(x+1)(x+2)}}{2x+3}}[/math]. It follows that this limit is 1, and therefore [math]\lim_{x\to\infty} {\frac{x\sqrt{x+2}}{\sqrt{x+1}}-x}=\frac{1}{1+1}=\frac{1}{2}[/math]. Edited September 28, 2016 by renerpho Link to post Share on other sites

Country Boy 70 Posted December 31, 2016 Share Posted December 31, 2016 Why use L'Hopital's rule? Mathematic's answer is much simpler. Link to post Share on other sites

ecoli 372 Posted December 31, 2016 Share Posted December 31, 2016 You can use wolframalpha to visualize the function, fyi: https://www.wolframalpha.com/input/?i=((x(sqrt(x%2B2)))%2Fsqrt(x%2B1))-x Link to post Share on other sites

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