List of top Mathematics Questions on Real Analysis asked in AP ECET BSc Mathematics

\[ \lim_{x\to0}\frac{e^{\frac{1}{x}}}{e^{\frac{1}{x}}+1}=\,\_ \]
For a Riemann integrability, the condition of continuity is
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}
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}
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}
If $f(x)=x,$ $x \in [0,3]$, and $P=\{0,1,2,3\}$ be a partition of [0,3], then $L(P,f)=$
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?
The function $f(x)=(2)^{\frac{1}{x}}$ is not continuous at}
If for $x\ne0$ and $f(x)=x^{3}sin\frac{1}{x}, f(0)=0$ then $f(x)$ is}
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 \(S_n=\displaystyle\sum_{k=1}^{n}\frac{n}{n^2+k}\), for \(n\in N\). Then the sequence \(\{S_n\}\) is ____.

The series \(\displaystyle\sum_{n=0}^{\infty}(2x)^n\) converges if ____.

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?
Which of the following statement is NOT CORRECT?
If \(f(x)=\sin\left(\dfrac{1}{x}\right)\), then \(f(x)\) is _ _ _ _ in \((0,2\pi]\).
Which of the following function is continuous but not uniformly continuous on the real line?
If \(f(x+y)=f(x)f(y),\ \forall x,y\), suppose \(f(3)=3\) and \(f'(0)=11\), then \(f'(3)=\)
The value of \(C\) of Rolle's theorem for the function \(f(x)=\sin x\) in \([0,\pi]\) is
If \(f(x)=1\) when \(x\in Q\) and \(f(x)=-1\) when \(x\in R-Q\), then
If \(f:[a,b]\to R\) is Riemann integrable, then
The values of infimum and supremum of the set \(S=\{x\in \mathbb{R}:x^3\leq x\}\) are
\(\displaystyle \lim_{n\to \infty}\frac{2^{n+1}+3^{n+1}}{2^n+3^n}=\)