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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 subdomains 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. Last edited by skipjack; October 4th, 2019 at 12:44 AM. 

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basic, exponent, limit 
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