A high number problem, the use of the pinch criterion LIM (1 + 2 ^ n + 3 ^ n) ^ (1 / N) where n tends to infinity, why is its size between 3 and 3 times 3 ^ (1 / N)

A high number problem, the use of the pinch criterion LIM (1 + 2 ^ n + 3 ^ n) ^ (1 / N) where n tends to infinity, why is its size between 3 and 3 times 3 ^ (1 / N)

The key to this problem is to expand and contract the expressions in LIM~
Let's look at the formula 1 + 2 ^ n + 3 ^ n in LIM brackets
Obviously, this formula is greater than 3 ^ n (without the first and second terms)
Then the limit must be greater than: LIM (3 ^ n) ^ (1 / N), and the value of LIM (3 ^ n) ^ (1 / N) is 3
The formula in LIM brackets is obviously less than 3 times 3 ^ n
Then, the limit must be less than: LIM (3 times 3 ^ n) ^ (1 / N), and as long as the landlord powers 3 ^ n and (1 / N) in brackets, we can get 3 times 3 ^ (1 / N)
So the result of the pinch rule is as above~