## Step 1: a. Analyzing \(n = o(2n)\)

We are given the statement \(n = o(2n)\) and must determine if it's true or false. Recall that \(f(n) = o(g(n))\) if and only if \(\lim_{n \to \infty} \frac{f(n)}{g(n)} = 0\). So let's find the limit for this statement.

## Step 2: a.1 Limit Calculation

We are given \(f(n) = n\) and \(g(n) = 2n\). Let's find the limit \(\lim_{n \to \infty} \frac{f(n)}{g(n)}\).

## Step 3: a.2 Limit Simplification

We have \(\lim_{n \to \infty} \frac{n}{2n}\). Both numerator and denominator have an n term, so we can simplify this expression to \(\lim_{n \to \infty} \frac{1}{2}\).

## Step 4: a.3 Limit Conclusion

Since the limit equals \(\frac{1}{2}\) and not 0, the statement \(n = o(2n)\) is FALSE.

## Step 5: b. Analyzing \(2n = o(n^2)\)

We are given the statement \(2n = o(n^2)\) and must determine if it's true or false. Let's find the limit for this statement.

## Step 6: b.1 Limit Calculation

We are given \(f(n) = 2n\) and \(g(n) = n^2\). Let's find the limit \(\lim_{n \to \infty} \frac{f(n)}{g(n)}\).

## Step 7: b.2 Limit Simplification

We have \(\lim_{n \to \infty} \frac{2n}{n^2}\). We can simplify the expression to \(\lim_{n \to \infty} \frac{2}{n}\).

## Step 8: b.3 Limit Conclusion

Since the limit equals 0, the statement \(2n = o(n^2)\) is TRUE.

## Step 9: c. Analyzing \(2^n = o(3^n)\)

We are given the statement \(2^n = o(3^n)\) and must determine if it's true or false. Let's find the limit for this statement.

## Step 10: c.1 Limit Calculation

We are given \(f(n) = 2^n\) and \(g(n) = 3^n\). Let's find the limit \(\lim_{n \to \infty} \frac{f(n)}{g(n)}\).

## Step 11: c.2 Limit Simplification

We have \(\lim_{n \to \infty} \frac{2^n}{3^n}\). We can simplify the expression to \(\lim_{n \to \infty} (\frac{2}{3})^n\).

## Step 12: c.3 Limit Conclusion

Since the limit equals 0, the statement \(2^n = o(3^n)\) is TRUE.

## Step 13: d. Analyzing \(1 = o(\log{n})\)

We are given the statement \(1 = o(\log{n})\) and must determine if it's true or false. Let's find the limit for this statement.

## Step 14: d.1 Limit Calculation

We are given \(f(n) = 1\) and \(g(n) = \log{n}\). Let's find the limit \(\lim_{n \to \infty} \frac{f(n)}{g(n)}\).

## Step 15: d.2 Limit Simplification

We have \(\lim_{n \to \infty} \frac{1}{\log{n}}\).

## Step 16: d.3 Limit Conclusion

Since the limit equals 0, the statement \(1 = o(\log{n})\) is TRUE.

## Step 17: e. Analyzing \(n = o(\log{n})\)

We are given the statement \(n = o(\log{n})\) and must determine if it's true or false. Let's find the limit for this statement.

## Step 18: e.1 Limit Calculation

We are given \(f(n) = n\) and \(g(n) = \log{n}\). Let's find the limit \(\lim_{n \to \infty} \frac{f(n)}{g(n)}\).

## Step 19: e.2 Limit Simplification

We have \(\lim_{n \to \infty} \frac{n}{\log{n}}\).

## Step 20: e.3 Limit Conclusion

Since the limit doesn't equal 0, the statement \(n = o(\log{n})\) is FALSE.

## Step 21: f. Analyzing \(1 = o(\frac{1}{n})\)

We are given the statement \(1 = o(\frac{1}{n})\) and must determine if it's true or false. Let's find the limit for this statement.

## Step 22: f.1 Limit Calculation

We are given \(f(n) = 1\) and \(g(n) = \frac{1}{n}\). Let's find the limit \(\lim_{n \to \infty} \frac{f(n)}{g(n)}\).

## Step 23: f.2 Limit Simplification

We have \(\lim_{n \to \infty} \frac{1}{\frac{1}{n}}\). We can simplify the expression to \(\lim_{n \to \infty} n\).

## Step 24: f.3 Limit Conclusion

Since the limit doesn't equal 0, the statement \(1 = o(\frac{1}{n})\) is FALSE.
In conclusion:
a. \(n = o(2n)\) is FALSE.
b. \(2n = o(n^2)\) is TRUE.
c. \(2^n = o(3^n)\) is TRUE.
d. \(1 = o(\log{n})\) is TRUE.
e. \(n = o(\log{n})\) is FALSE.
f. \(1 = o(\frac{1}{n})\) is FALSE.