MHB Limit to Infinity: Is $$\lim_{{x}\to{\infty}} \sqrt{x^2 + 2x + 1} - x = 1?$$

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The limit $$\lim_{{x}\to{\infty}} \sqrt{x^2 + 2x + 1} - x$$ is indeed equal to 1. Initial calculations may suggest it equals 0 due to the behavior of infinity, but this approach overlooks the need to simplify the expression. By rewriting the radicand as a perfect square, the limit can be accurately evaluated. The correct method reveals that the limit approaches 1 as x approaches infinity. Thus, the assertion that the limit equals 1 is confirmed.
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I have in my notes that $$\lim_{{x}\to{\infty}} \sqrt{x^2 + 2x + 1} - x = 1$$

Is this right? When I calculate it, I get 0, because the square root of infinity is infinity and then I subtract infinity which is 0.
 
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You can't legitimately subtract infinities...but what you can do is rewrite the radicand as a square...then the desired result is forthcoming. :D
 

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