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Problem 2

# Answer each part TRUE or FALSE. a. $$n=o(2 n)$$. A $$\mathbf{d .} \quad 1=o(n)$$. b. $$2 n=o\left(n^{2}\right)$$. e. $$n=o(\log n)$$. A c. $$2^{n}=o\left(3^{n}\right)$$. f. $$1=o(1 / n)$$.

Expert verified
a. FALSE b. TRUE c. TRUE d. TRUE e. FALSE f. FALSE
See the step by step solution

## 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.

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