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Calculuate $\displaystyle\lim_{x\to 0^+}x\ln{x}$.

Calculate $\displaystyle\lim_{x\to 0^+}x^x$.

Compute $$\lim_{n\to\infty}n^2\int_0^{\frac{1}{n}}x^{2018x+1}dx$$

There is a unique positive real number $a$ such that the tangent line to $y = x^2 + 1$ at $x = a$ goes through the origin. Compute $a$.

Moor has $\$1000$, and he is playing a gambling game. He gets to pick a number k between$0$and$1$(inclusive). A fair coin is then flipped. If the coin comes up heads, Moor is given$5000k$additional dollars. Otherwise, Moor loses$1000k$dollars. Moor’s happiness is equal to the log of the amount of money that he has after this gambling game. Find the value of k that Moor should select to maximize his expected happiness. Compute $$\lim_{x\to\infty}\left[x-x^2\ln\left(\frac{1+x}{x}\right)\right]$$ Find the value of$\displaystyle\lim_{n\to\infty}\sin^2\left(\pi\sqrt{n^2+n}\right)$. Compute the value of $$\lim_{x\to\pi}\frac{\ln(2+\cos{x})}{\left(3^{\sin{x}}-1\right)^2}$$ Compute the value of $$\lim_{n\to\infty}n^2\left(1-\cos\frac{\pi}{n}\right)$$ If water is poured into a right cone whose height is$H$cm and base's radius is$R$cm at a speed of$Acm^3\$ per second, what is the speed the water is rising when the depth of water is half of the cone's height?