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October 3rd, 2019, 08:47 PM   #1
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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 $\displaystyle 0< r \neq 1$ .
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October 3rd, 2019, 10:44 PM   #2
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Step 1: Break it into two sub-domains and simplify
\(\displaystyle \lim_{k \rightarrow \infty} r^k = \lim_{k \rightarrow \infty} r^{k^2} - r^{2k^2} ~;~ 0<r<1 \)

\(\displaystyle \lim_{k \rightarrow \infty} r^k = \lim_{k \rightarrow \infty} r^{2k^2} - r^{k^2} ~;~ r>1 \)

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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