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Mathematics
General Math
Proving Limits at Infinity: Can You Help Me with These Two Limits?
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[QUOTE="Opalg, post: 6778918, member: 703375"] Hi Goody, and welcome to MHB! To prove that [MATH]\lim_{x\to\infty}\frac{x-1}{x+2} = 1[/MATH], you have to show that, given $\varepsilon > 0$, you can find $N$ such that [MATH]\left|\frac{x-1}{x+2} - 1\right| < \varepsilon[/MATH] whenever $x>N$. So, first you should simplify [MATH]\left|\frac{x-1}{x+2} - 1\right|[/MATH]. Then you should see how large $x$ has to be in order to make that expression less than $\varepsilon$. To prove that [MATH]\lim_{x\to-1}\frac{-1}{(x+1)^2} = -\infty[/MATH], you have to show that, given $M$, you can find $\delta>0$ such that [MATH]\frac{-1}{(x+1)^2} < -M[/MATH] whenever $|x+1| < \delta$. That is actually an easier calculation than the first one, so you might want to try that one first. [/QUOTE]
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Proving Limits at Infinity: Can You Help Me with These Two Limits?
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