$g$ is a twice differentiable function over the positive reals such that \begin{align}g(x)+2x^3g^\prime(x)+x^4g^{\prime\prime}(x)&= 0 \qquad\text{ for all positive reals } x\\\lim_{x\to\infty}xg(x)&=1\end{align} Find the real number $\alpha>1$ such that $g(\alpha)=1/2$.

Find the following limit. \[\lim_{n\to\infty} \frac{1}{n} \log \left(\sum_{k=2}^{2^n} k^{1/n^2} \right)\]

Let $a,\ b$ be positive real numbers with $a<b$. Define the definite integrals $I_1,\ I_2,\ I_3$ by $I_1=\int_a^b \sin\ (x^2)\ dx,\ I_2=\int_a^b \frac{\cos\ (x^2)}{x^2}\ dx,\ I_3=\int_a^b \frac{\sin\ (x^2)}{x^4}\ dx$. (1) Find the value of $I_1+\frac 12I_2$ in terms of $a,\ b$. (2) Find the value of $I_2-\frac 32I_3$ in terms of $a,\ b$. (3) For a positive integer $n$, define $K_n=\int_{\sqrt{2n\pi}}^{\sqrt{2(n+1)\pi}} \sin\ (x^2)\ dx+\frac 34\int_{\sqrt{2n\pi}}^{\sqrt{2(n+1)\pi}}\frac{\sin\ (x^2)}{x^4}\ dx$. Find the value of $\lim_{n\to\infty} 2n\pi \sqrt{2n\pi} K_n$. [i]2011 Tokyo University of Science entrance exam/Information Sciences, Applied Chemistry, Mechanical Enginerring, Civil Enginerring[/i]

Real number $C > 1$ is given. Sequence of positive real numbers $a_1, a_2, a_3, \ldots$, in which $a_1=1$ and $a_2=2$, satisfy the conditions \[a_{mn}=a_ma_n, \] \[a_{m+n} \leq C(a_m + a_n),\] for $m, n = 1, 2, 3, \ldots$. Prove that $a_n = n$ for $n=1, 2, 3, \ldots$.

Let be the sequence $ \left( a_n \right)_{n\ge 0} $ of positive real numbers defined by $$ a_n=1+\frac{a_{n-1}}{n} ,\quad\forall n\ge 1. $$ Calculate $ \lim_{n\to\infty } a_n ^n . $ [i]Florian Dumitrel[/i]

A function $ f(x)$ is [i]periodic[/i] if there is a positive number $ p$ such that $ f(x\plus{}p) \equal{} f(x)$ for all $ x$. For example, $ \sin x$ is periodic with period $ 2 \pi$. Is the function $ \sin(x^2)$ periodic? Prove your assertion.

[b]a)[/b] Prove that $ \lim_{x\to\infty } \sqrt{x}\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x}=1. $ [b]b)[/b] Show that $ \lim_{x\to\infty } \left( -\left\lfloor\sqrt{x}\right\rfloor +x\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x} \right) =\frac{-1}{2} $ [b]c)[/b] What about $ \lim_{x\to\infty } \left( -\sqrt{x} +x\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x} \right) ? $

Let $\alpha$ be a solution satisfying the equation $|x|=e^{-x}.$ Let $I_n=\int_0^{\alpha} (xe^{-nx}+\alpha x^{n-1})dx\ (n=1,\ 2,\ \cdots).$ Find $\lim_{n\to\infty} n^2I_n.$

In the $xy$-plane, for $a>1$ denote by $S(a)$ the area of the figure bounded by the curve $y=(a-x)\ln x$ and the $x$-axis. Find the value of integer $n$ for which $\lim_{a\rightarrow \infty} \frac{S(a)}{a^n\ln a}$ is non-zero real number.

Let $(a_n )$ be a sequence of real numbers satisfying the inequalities $$ 0 \leq a_k \leq 100a_n \;\; \text{for} \;\, n \leq k \leq 2n \;\; \text{and} \;\; n=1,2,\ldots,$$ and such that the series $$\sum_{n=0}^{\infty} a_n $$ converges. Prove that $$\lim_{n\to \infty} n a_n = 0.$$

$ \lim_{n\to\infty } \int_{\pi }^{2\pi } \frac{|\sin (nx) +\cos (nx)|}{ x} dx ? $ [i]Gabriela Boeriu[/i]

Two positive valued sequences $\{ a_{n}\}$ and $\{ b_{n}\}$ satisfy: (a): $a_{0}=1 \geq a_{1}$, $a_{n}(b_{n+1}+b_{n-1})=a_{n-1}b_{n-1}+a_{n+1}b_{n+1}$, $n \geq 1$. (b): $\sum_{i=1}^{n}b_{i}\leq n^{\frac{3}{2}}$, $n \geq 1$. Find the general term of $\{ a_{n}\}$.

Let be a sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1>0 $ and satisfying the equality $$ a_n=\sqrt{a_{n+1} -\sqrt{a_{n+1} +a_n}} , $$ for all natural numbers $ n. $ [b]a)[/b] Find a recurrence relation between two consecutive elements of $ \left( a_n \right)_{n\ge 1} . $ [b]b)[/b] Prove that $ \lim_{n\to\infty } \frac{\ln\ln a_n}{n} =\ln 2. $

Let $f(x)$ be a real-valued function satisfying $af(x)+bf(-x)=px^2+qx+r$. $a$ and $b$ are distinct real numbers and $p,q,r$ are non-zero real numbers. Then $f(x)=0$ will have real solutions when (A)$\left(\frac{a+b}{a-b}\right)\leq\frac{q^2}{4pr}$ (B)$\left(\frac{a+b}{a-b}\right)\leq\frac{4pr}{q^2}$ (C)$\left(\frac{a+b}{a-b}\right)\geq\frac{q^2}{4pr}$ (D)$\left(\frac{a+b}{a-b}\right)\geq\frac{4pr}{q^2}$