Question

EVALUATE sqrt{X ^{2}+2}/ X-1 AS X GOES TO INFINITY

Answer 1

\[\lim_{x \rightarrow \infty} \sqrt{x^{2}+2}/(x-1)\]
Factor out an x^2 in the numerator and an x in the denominator. This gets you the following:
\[\lim_{x \rightarrow \infty}\sqrt{x^{2}(1+(2/x^{2}))}/x(1-(1/x))\]
\[\lim_{x \rightarrow \infty}x\sqrt{(1+(2/x^{2}))}/x(1-(1/x))\]
\[\lim_{x \rightarrow \infty}\sqrt{(1+(2/x^{2}))}/(1-(1/x))\]
Notice that 2/x^2 and 1/x go to zero as x -> infinity because as the denominator of a fraction gets larger, the value of the fraction gets smaller. So,
\[\lim_{x \rightarrow \infty}\sqrt{(1+(2/x^{2}))}/(1-(1/x))0 =\sqrt{(1+0)}/(1-0)\]
And the square root of 1 over 1 equals 1 so,
\[\lim_{x \rightarrow \infty} \sqrt{x^{2}+2}/(x-1) = 1\]

Answer 2

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