Write out an explanatory paragraph for the following limits that include ∞ {\displaystyle \infty } . Remember that you will have to change any comparison of magnitude between a real number and ∞ {\displaystyle \infty } to a different phrase. In the second case, you will have to work out for yourself what the formula means. 1. lim x → ∞ 1 x 2 = 0 {\displaystyle \lim _{x\to \infty }{\frac {1}{x^{2}}}=0} This formula says that I can make the values of 1 x 2 {\displaystyle {\frac {1}{x^{2}}}} as close as I would like to 0, so long as I make x sufficiently large. This formula says that I can make the values of 1 x 2 {\displaystyle {\frac {1}{x^{2}}}} as close as I would like to 0, so long as I make x sufficiently large. 2. ∑ n = 0 ∞ 2 − n = 1 + 1 2 + 1 4 + 1 8 + ⋯ = 2 {\displaystyle \sum _{n=0}^{\infty }2^{-n}=1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots =2} This formula says that you can make the sum ∑ n = 0 i 2 − n {\displaystyle \sum _{n=0}^{i}2^{-n}} as close as you would like to 2 by making i {\displaystyle i} sufficiently large. This formula says that you can make the sum ∑ n = 0 i 2 − n {\displaystyle \sum _{n=0}^{i}2^{-n}} as close as you would like to 2 by making i {\displaystyle i} sufficiently large.
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