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 October 3rd, 2019, 08:47 PM #1 Senior Member   Joined: Dec 2015 From: somewhere Posts: 734 Thanks: 98 Basic limit with exponent How can I prove the equality ? $\displaystyle \lim_{k\rightarrow \infty} r^{k}=\lim_{k\rightarrow \infty } |r^{2}-r|^{k^2 }\:$ ; for $\displaystyle 0< r \neq 1$ .
 October 3rd, 2019, 10:44 PM #2 Senior Member   Joined: Jun 2019 From: USA Posts: 310 Thanks: 162 Step 1: Break it into two sub-domains and simplify $$\displaystyle \lim_{k \rightarrow \infty} r^k = \lim_{k \rightarrow \infty} r^{k^2} - r^{2k^2} ~;~ 01$$ Second case, both sides are unbounded towards infinity, thus equal. First case, let $$s = r^{-1} \rightarrow s>1$$ $$\displaystyle \rightarrow \lim \frac{1}{s^k} = \lim \frac{1}{s^{k^2}} - \frac{1}{s^{2k^2}} = \lim \frac{s^{k^2}-1}{s^{2k^2}}$$ Both sides go to zero, thus equal. Thanks from tahirimanov19 and idontknow Last edited by skipjack; October 4th, 2019 at 12:44 AM.

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