lim n infty (n^ 2 cos( 6 n^ 2 + pi 2 ))=
not sure what "pi 2" means, but as n gets large, it's irrelevant, and you just have cos(6n^2)
Since cosine oscillates, the limit is undefined.
To find the limit of the given expression as n approaches infinity, we can use some basic concepts of limits. Let's break down the steps to find the limit of the expression:
Step 1: Rewrite the expression in proper mathematical notation.
The given expression is lim(n → ∞) (n^2 * cos(6n^2 + π/2)).
Step 2: Identify the dominant term.
In this expression, the dominant term is n^2, as it grows much faster than the trigonometric term inside the cosine function.
Step 3: Evaluate the limiting behavior of the dominant term.
Taking the limit of the dominant term will give us an idea of how the expression behaves as n approaches infinity. The limit of n^2 as n approaches infinity is infinity (or +∞).
Step 4: Evaluate the limiting behavior of the trigonometric function.
The cosine function oscillates between -1 and 1 as its argument increases. The argument, 6n^2 + π/2, is an increasing function as n approaches infinity, but it does not approach any specific value. Therefore, the cosine function will oscillate as well.
Step 5: Multiply the limiting behavior of the dominant term and the trigonometric function.
Since the cosine function oscillates between -1 and 1, multiplying it by the growing term n^2 will produce a result that alternates between -∞ and +∞.
Step 6: Finalize the answer.
Combining the results from Step 4 and Step 5, we conclude that lim(n → ∞) (n^2 * cos(6n^2 + π/2)) does not exist. The expression oscillates between infinite positive and negative values indefinitely as n approaches infinity.