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pshichka [43]
3 years ago
12

Compute the limit of the following as x approaches infinity.

Mathematics
1 answer:
Nonamiya [84]3 years ago
3 0

Yes, that's the correct limit.

\sqrt{x^2+1}-x=\dfrac1{\sqrt{x^2+1}+x}=\dfrac1{\sqrt{x^2}\sqrt{1+\frac1{x^2}}+x}

\sqrt{x^2}=|x|, but since x\to\infty, we are considering x>0, for which |x|=x. Then

\displaystyle\lim_{x\to\infty}\sqrt{x^2+1}-x=\lim_{x\to\infty}\frac1{x\left(\sqrt{1+\frac1{x^2}}+1\right)}

and both \dfrac1x and \dfrac1{x^2} vanish as x\to\infty, making the overall limit 0.

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