List of top Mathematics Questions on Real Analysis

\[ \lim_{x\to0}\frac{e^{\frac{1}{x}}}{e^{\frac{1}{x}}+1}=\,\_ \]

For a Riemann integrability, the condition of continuity is

The value of C of the Cauchy's mean value theorem for the function $f(x)=e^{x}$ and $g(x)=e^{-x}$ defined on $[a,b]$ is}

If $f(x)=x,$ $x \in [0,3]$, and $P=\{0,1,2,3\}$ be a partition of [0,3], then $L(P,f)=$

Let $f(x)=\begin{cases}x^{p}cos\frac{1}{x},x\ne0\\ 0,x=0\end{cases}$ Then $f(x)$ is differentiable at $x=0$ if}

The function f is defined by $f(x)=\begin{cases}x^{2}+3x+a,& if~x\le1\\ bx+2,& if~x>1\end{cases}$ is derivable for every x. Then the value of a is}

If the function $f(x)$ is bounded and integrable on $[a,b]$ such that $f(x) \ge 0$ $\forall x \in [a,b]$, $b \ge a$ then $\int_{a}^{b} f(x) dx$ is __}

The least upper bound of the set $\{\frac{3n+2}{2n+1}/n\in N\}$ where N is set of natural numbers is}

Which of the following is a convergent series?

If for $x\ne0$ and $f(x)=x^{3}sin\frac{1}{x}, f(0)=0$ then $f(x)$ is}

The function $f(x)=(2)^{\frac{1}{x}}$ is not continuous at}

If $f(x+y)=f(x)+f(y)+x^{2}y+yx^{2},\forall x,y\in R,$ and $\lim_{x\to0}\frac{f(x)}{x}=1$. Then $f'(3)$ is equal to}

A non-empty sub-set of real numbers which is bounded below has

Let \( f: \mathbb{R} \to \mathbb{R} \) be the function given by \[ f(x) = \begin{cases} \cos x, & \text{if } x \in \mathbb{Q} \\ 0, & \text{if } x \notin \mathbb{Q} \end{cases} \]
Which of the following statements is/are TRUE?

Find the radius of convergence of the series
\[ \sum_{n=0}^{\infty} \frac{\binom{n}{6}^2}{(2n)!} x^n \]

Find the number of automorphisms of the cyclic group $\mathbb{Z}_n$ for $n = 30$.

Determine whether the sequence \[ a_n = 1 - (-1)^n + \frac{1}{n} \] is convergent or divergent.

Let $G = P(N)$, where the operation is
\[ A \Delta B = A \cup B - A \cap B \] Which of the following is true?

Find the radius of convergence of the series \[ \sum_{n=0}^{\infty} \frac{(n!)^2}{(2n)!}\, x^n. \]

Which of the following statement is NOT CORRECT?

If \(f,g\) are bounded functions defined on \([a,b]\) and let \(P\) be any partition of \([a,b]\), then which of the following is NOT TRUE?

If \(f(x)=1\) when \(x\in Q\) and \(f(x)=-1\) when \(x\in R-Q\), then

The value of \(C\) of Rolle's theorem for the function \(f(x)=\sin x\) in \([0,\pi]\) is