Prove thatlimn→∞(4n)1/n = 1
Solution
Sure, here is the step-by-step proof:
Step 1: Rewrite the expression using the properties of exponents
The expression (4n)^(1/n) can be rewritten as 4^(n/n) = 4.
Step 2: Apply the limit
The limit as n approaches infinity of a constant (in this case, 4) is just the constant itself. So, lim(n→∞) 4 = 4.
However, there seems to be a mistake in your question. The limit of (4n)^(1/n) as n approaches infinity is not 1, it's 4. If you meant the limit of (n^4)^(1/n), then the limit would be 1.
Here's how you can prove that:
Step 1: Rewrite the expression using the properties of exponents
The expression (n^4)^(1/n) can be rewritten as n^(4/n).
Step 2: Apply the limit
As n approaches infinity, the expression 4/n approaches 0. So, the expression becomes n^0, which is 1.
So, lim(n→∞) (n^4)^(1/n) = 1.
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