evaluate the limit

Question

lt n-infinity ((q^n) + (p^n))^(1/n)

Answer 1

If  q < p   the limit becomes

lim_n→∞ p (1 + (q/p)ⁿ )^1/n   = p lim_n→∞ (1 + (q/p)ⁿ )^1/n   =  p

 

This because it can be shown that lim_n→∞ ( 1 + zⁿ )^1/n   =1    (for z <  1)

A little experimentation with a calculator can verify this.  The analytic approach involves taking the log and then using 

L' Hospital's rule.  

 

If  q > p   then the limit is q.    

If q = p,   the limit is  p (which is equal to q)