List of top Mathematics Questions on Limit Theorems

Let \( a_n = \sqrt{n}, n \geq 1 \), and let \[ s_n = a_1 + a_2 + \cdots + a_n. \text{ Then} \] \[ \lim_{n \to \infty} \left( \frac{a_n}{s_n} \right) \left( -\ln \left( 1 - \frac{a_n}{s_n} \right) \right) = \text{............}. \]

Evaluate the limit \[ \lim_{n \to \infty} \frac{1}{\sqrt{n}} \left( \frac{1}{\sqrt{3} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{9}} + \cdots + \frac{1}{\sqrt{3n} + \sqrt{3n + 3}} \right). \]

Let \[ M = \begin{bmatrix} 1 & 1 \\ 2 & 4 \end{bmatrix} \quad \text{and} \quad x = \begin{bmatrix} 3 \\ 4 \end{bmatrix}. \] Then \[ \lim_{n \to \infty} M^n x \]

Let \( f(x) = \frac{x + |x|(1 + x)}{x} \sin \left( \frac{1}{x} \right) \), for \( x \neq 0 \). Write \( L = \lim_{x \to 0^-} f(x) \) and \( R = \lim_{x \to 0^+} f(x) \). Then which one of the following is TRUE?

Find the limit: \[ \lim_{n \to \infty} \sum_{k=1}^{n} \sin \left( \frac{\pi}{2} + \frac{5\pi}{2} \cdot \frac{k}{n} \right) = \]