### Limit of Integration (TIFR 2014 problem 7)

Question: Let \(f_n(x); n\ge 1\) be a sequence of continuous nonnegative functions on \(0,1\) such that \( \lim_{n\to\infty} \int_{0}^{1}f_n(x)dx = …

Question: Let \(f_n(x); n\ge 1\) be a sequence of continuous nonnegative functions on \(0,1\) such that \( \lim_{n\to\infty} \int_{0}^{1}f_n(x)dx = …

Question: Let \(f:0,1 \to \mathbb{R} \) be a continuous function. Which of the following statement is always true? A. \( …

Question: Let \(a_n= (n+1)^{100}e^{-\sqrt{n}}\) for \(n \ge 1\). Then the sequence \(a_n\) is A. unbounded B. bounded but not convergent …

Question: Let \(f:0,\infty)\to \mathbb{R}\) be defined by \( f(x)=x^{2/3}logx\) for \(x>0 \) \(f(x)=0\) if \(x=0\) Then A. f is discontinuous …

Question: Let \(f:\mathbb{R}\to \mathbb{R}\) be a differentiable function such that \(\lim_{x\to \infty} f'(x)=1\) then A. f is bounded B. f …

Question: (MCQ) Let \(f:\mathbb{R}\to \mathbb{R}\) be a continuous bounded function. Then A. f has to be uniformly continuous B. there …

Question: (MCQ) Let A,B,C be three subsets of \(\mathbb{R}\) The negation of the following statement: For every \(\epsilon >1\), there …

Question: True/False? The series \(1-\frac{1}{\sqrt2}+\frac{1}{\sqrt3}-\frac{1}{\sqrt4}+…\) is divergent. Hint: Recall the alternating series test (or the Leibniz test) Discussion: Let \(a_n=\frac{1}{\sqrt{n}}\). …

Question: True/False? If \(A\) is a complex nxn matrix with \(A^2=A\), then rank\(A\)=trace\(A\). Hint: What are the eigenvalues of \(A\)? …

Question: True/False? Let \(V\) be the vector space of polynomials with real coefficients in variable \(t\) of degree \( \le …