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Limits - square root

  • #1
Rectifier
Gold Member
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4
The problem
$$ \lim_{x \rightarrow \infty} \left( \sqrt{x+1} - \sqrt{x} \right) $$

The attempt
## \left( \sqrt{x+1} - \sqrt{x} \right) = \frac{\left( \sqrt{x+1} - \sqrt{x} \right)\left( \sqrt{x+1} + \sqrt{x} \right) }{\left( \sqrt{x+1} + \sqrt{x} \right) } = \frac{x+1 - x }{\left( \sqrt{x+1} + \sqrt{x} \right) } = \frac{1 }{x \left( \sqrt{\frac{1}{x}+\frac{1}{x^2}} + \sqrt{\frac{1}{x}} \right) } = \frac{1}{x} \frac{1 }{ \sqrt{\frac{1}{x}+\frac{1}{x^2}} + \sqrt{\frac{1}{x}} } \\ \rightarrow 0 \cdot \frac{1}{0 + 0} ## for ## x \rightarrow \infty ## division with 0 is undefined. Can somone help me calculate this limit.

And no, I cant use l'Hopitals rule so please don't mention it in this thread.
 

Answers and Replies

  • #2
Math_QED
Science Advisor
Homework Helper
2019 Award
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The problem
$$ \lim_{x \rightarrow \infty} \left( \sqrt{x+1} - \sqrt{x} \right) $$

The attempt
## \left( \sqrt{x+1} - \sqrt{x} \right) = \frac{\left( \sqrt{x+1} - \sqrt{x} \right)\left( \sqrt{x+1} + \sqrt{x} \right) }{\left( \sqrt{x+1} + \sqrt{x} \right) } = \frac{x+1 - x }{\left( \sqrt{x+1} + \sqrt{x} \right) } = \frac{1 }{x \left( \sqrt{\frac{1}{x}+\frac{1}{x^2}} + \sqrt{\frac{1}{x}} \right) } = \frac{1}{x} \frac{1 }{ \sqrt{\frac{1}{x}+\frac{1}{x^2}} + \sqrt{\frac{1}{x}} } \\ \rightarrow 0 \cdot \frac{1}{0 + 0} ## for ## x \rightarrow \infty ## division with 0 is undefined. Can somone help me calculate this limit.

And no, I cant use l'Hopitals rule so please don't mention it in this thread.
## \left( \sqrt{x+1} - \sqrt{x} \right) = \frac{\left( \sqrt{x+1} - \sqrt{x} \right)\left( \sqrt{x+1} + \sqrt{x} \right) }{\left( \sqrt{x+1} + \sqrt{x} \right) } = \frac{1 }{\left( \sqrt{x+1} + \sqrt{x} \right) } ##.

Until here, what you do leads to the answer. When you fill in infinity right now, you don't get an undetermined form, so the answer is ...?
 
  • #3
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
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The problem
$$ \lim_{x \rightarrow \infty} \left( \sqrt{x+1} - \sqrt{x} \right) $$

The attempt
## \left( \sqrt{x+1} - \sqrt{x} \right) = \frac{\left( \sqrt{x+1} - \sqrt{x} \right)\left( \sqrt{x+1} + \sqrt{x} \right) }{\left( \sqrt{x+1} + \sqrt{x} \right) } = \frac{x+1 - x }{\left( \sqrt{x+1} + \sqrt{x} \right) } = \frac{1 }{x \left( \sqrt{\frac{1}{x}+\frac{1}{x^2}} + \sqrt{\frac{1}{x}} \right) } = \frac{1}{x} \frac{1 }{ \sqrt{\frac{1}{x}+\frac{1}{x^2}} + \sqrt{\frac{1}{x}} } \\ \rightarrow 0 \cdot \frac{1}{0 + 0} ## for ## x \rightarrow \infty ## division with 0 is undefined. Can somone help me calculate this limit.

And no, I cant use l'Hopitals rule so please don't mention it in this thread.
Note that
[tex] \frac{x+1-x}{\sqrt{x}+\sqrt{x+1}} = \frac{1}{\sqrt{x}+\sqrt{x+1}} [/tex]
In the second form, what happens to the numerator when ##x \to +\infty##? What happens to the denominator?

What dictator forbids you from using l'Hospital's rule?
 
  • #4
33,635
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What dictator forbids you from using l'Hospital's rule?
I haven't worked this problem, but sometimes L'Hopital's rule doesn't get you anywhere. I've seen similar problems where two applications of L'H gets you right back to where you started.
 

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