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 Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion

 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. Tags basic, exponent, limit Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post idontknow Real Analysis 8 August 1st, 2018 01:52 PM nbg273 Calculus 7 February 18th, 2017 01:48 AM DerGiLLster Calculus 1 September 16th, 2015 06:44 PM Tutu Calculus 4 June 26th, 2012 04:08 AM sjeddie Calculus 3 August 28th, 2008 07:45 AM

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