Let $$f:\R \to \R$$ be a differential function such that f and its derivative f' have no common zeros in [0,1] . which one of the following statements is correct ?​

$$\lim_{n \to \infty} \frac{((n+1)(n+2) \cdots (n+n))^{1/n}}{n}.$$​

Suppose $$f:[0,1]\times[0,1] \to (0,1)\times (0,1)$$ is a continuous,

non constant function. Which of the following statements 

are NOT true ?

Let f be a non constant polynomial of degree k and let g:$$R\to R$$ be a bounded continuous function

Which of the following statements are necessarily true? 

​Let S be a dense subset of R and,

Which of the following statements is true ?

​$$\text{Let } (X, d) \text{ be a metric space and let } f: X \to X \text{ be a function such that:}\\d(f(x), f(y)) \leq d(x, y), \quad \text{for every } x, y \in X.\\\text{Which of the following statements is necessarily true?}$$​​

Let X be a non empty finite set and,

​$$\text{Given } f, g \text{ are continuous functions on } [0, 1] \text{ such that } \\f(0) = f(1) = 0, \; g(0) = g(1) = 1, \; \text{and } f(1/2) > g(1/2). \\\text{Which of the following statements is true?}$$​​

​​

$$\lim_{n \to \infty} \frac{((n+1)(n+2) \cdots (n+n))^{1/n}}{n}.$$​