Sure, let's solve this step by step:
We are asked to find the limit of the sequence a_n = (3 + n^57)^(1/5) as n approaches infinity.
Step 1:
To solve this, we begin by analyzing the behavior of the function for large values of n.
Step 2:
For very large values of n, the 3 becomes insignificant as n^57 dominates the equation.
Step 3:
Analysing the given sequence (3 + n^57)^(1/5), the term n^57 becomes dominant as n approaches infinity, making the constant term 3 negligible.
Step 4:
Therefore, the sequence behaves like (n^57)^(1/5) for large n values.
Step 5:
This simplifies to n^(57/5) which is n^11.4 as the sequence approaches infinity.
Step 6:
Since n^11.4 becomes infinitely large as n approaches infinity, the limit of the sequence is also infinite.
Therefore, the limit of the sequence a_n as n approaches infinity is infinity.
This approach is called limiting behavior analysis and it involves analyzing the behavior of a function as it approaches a certain point.