How would I go about finding the limit of this sequence:
(e^(2n)+6n)^(1/n)
Please explain the steps, it'd be super helpful! Thanks!
To find the limit of the sequence (e^(2n)+6n)^(1/n), we will use the property of the exponential function, specifically the limit of the form e^(f(n)). Here are the step-by-step instructions:
Step 1: Start with the given sequence (e^(2n)+6n)^(1/n).
Step 2: Rewrite the expression inside the parentheses using exponent rules. The expression becomes (e^(2n))/(n^(1/n)) + (6n)^(1/n).
Step 3: Take the limit as n approaches infinity of each term separately.
For the first term (e^(2n))/(n^(1/n)), we can rewrite it as (e^(2n))/(e^(ln(n^(1/n)))).
Since e^(ln(x)) = x, the expression simplifies to e^(2n - ln(n^(1/n))).
Now, let's focus on the exponent: 2n - ln(n^(1/n)).
We apply the property that ln(a^b) = b * ln(a).
The exponent can be rewritten as 2n - (1/n) * ln(n).
As n approaches infinity, the term (1/n) * ln(n) approaches 0, since the logarithmic function grows slower than any exponential function.
Therefore, the expression simplifies to 2n.
Step 4: Taking the limit of e^(2n) as n approaches infinity, we can use the property that e^(f(n)) approaches infinity as n approaches infinity for any positive f(n).
Thus, the limit of e^(2n) as n approaches infinity is also infinity.
Step 5: Finally, consider the second term (6n)^(1/n). The limit as n approaches infinity can be evaluated by recognizing that it is a power expression with base 6n and exponent 1/n.
Using the property that a^(1/n) approaches 1 as n approaches infinity for any positive constant a, we can conclude that the limit of (6n)^(1/n) as n approaches infinity is 1.
Step 6: Reassemble the limits from Steps 4 and 5. The ultimate limit is infinity + 1.
Therefore, the limit of the given sequence is infinity.
Keep in mind that these steps assume certain mathematical properties and are subject to mathematical rigor. These steps demonstrate the thought process behind finding the limit, but it is always advisable to verify your answer with further mathematical analysis or consult a professional mathematician if needed.