Find
limn→∞(n√(7n+n)−17)n7n−n7
I know that it can be done with using the squeeze theorem but I cannot find a proper upper bound limit
Answer
You may write
(n√(7n+n)−17)n7n−n7=(7(1+n7n)1/n−17)n7n(1−n77n)=7n(1+O(17n)−149)n7n(1−n77n)=(1+O(17n)−149)n(1−n77n)=(4849)n(1+O(17n))n1−n77n=(4849)n(1+O(n7n))1−n77n∼(4849)n
and the desired limit is equal to 0.
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