List of top Mathematics Questions on Limit and Continuity asked in IIT JAM MA

\[ \lim_{x\to 0}\left(\frac{1}{x}-\frac{1}{\sin x}+e^{\frac{1-\cos x}{x}}\right) = \underline{} \] rounded off to one decimal place.

Let \( f, g : \mathbb{R} \to \mathbb{R} \) be two functions defined by \[ f(x) = \begin{cases} x |x| \sin \frac{1}{x} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] and \[ g(x) = \begin{cases} x^2 \sin \frac{1}{x} + x \cos \frac{1}{x} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] Then, which one of the following is TRUE?

Let \( f, g : \mathbb{R} \to \mathbb{R} \) be two functions defined by \[ f(x) = \begin{cases} |x|^{1/8} \sin \tfrac{1}{|x|} \cos x & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] and \[ g(x) = \begin{cases} e^x \cos \tfrac{1}{x} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases} \] Then, which one of the following is TRUE?

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be defined by \[ f(x, y) = \begin{cases} \frac{(x^2 + \sin(xy))^2}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0), \\ 0 & \text{if } (x, y) = (0, 0). \end{cases} \] Then, which of the following is/are TRUE?

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be defined by \[ f(x, y) = \begin{cases} \frac{(x^2 - y^2)^2 xy}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0), \\ 0 & \text{if } (x, y) = (0, 0). \end{cases} \] Then, the value of \( \frac{\partial f}{\partial y}(0, 0) \) and \( \frac{\partial f}{\partial x}(0, 0) \) is equal to ........... (rounded off to two decimal places).

Let \( f(x) = 2x - \sin(x) \), for all \( x \in \mathbb{R} \). Let \( k \in \mathbb{N} \) be such that \[ \lim_{x \to 0} \left( \frac{1}{x} \sum_{i=1}^{k} i^2 f \left( \frac{x}{i} \right) \right) = 45. \] Then, the value of \( k \) is equal to ..............

The value of \[ \lim_{n \to \infty} 8n \left( \left( e^{\frac{1}{2n}} - 1 \right) \left( \sin \frac{1}{2n} + \cos \frac{1}{2n} \right) \right) \] is equal to ............... (rounded off to two decimal places).

Which of the following is FALSE?

Let

\[ f(x, y) = \begin{cases} \frac{|x|}{|x| + |y|} \sqrt{x^4 + y^2}, & (x, y) \neq (0, 0), \\ 0, & (x, y) = (0, 0). \end{cases} \]

Then at \( (0, 0) \),

For \( \beta \in \mathbb{R} \), define

\[ f(x, y) = \begin{cases} x^2 |x|^\beta y & \text{if } x \neq 0, \\ 0 & \text{if } x = 0. \end{cases} \]

Then, at \( (0, 0) \), the function \( f \) is

The greatest lower bound of the set \[ \left\{ \left( e^n + 2^n \right)^{\frac{1}{n}} : n \in \mathbb{N} \right\}, \] (round off to 2 decimal places) is .............

Let \( \{a_n\} \) be the sequence of real numbers such that \[ a_1 = 1 \quad \text{and} \quad a_{n+1} = a_n + a_n^2 \quad \text{for all } n \geq 1. \] Then

Let \( f : [0,1] \to \mathbb{R} \) be given by \[ f(x) = \frac{\left( \frac{1}{1+x^3} \right)^3 + \left( \frac{1}{1-x^3} \right)^3}{8(1+x)}. \] Then \( \max \{ f(x) : x \in [0,1] \} - \min \{ f(x) : x \in [0,1] \} \) is ...........

Let \( f(x) = (\ln x)^2, x>0 \). Then