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 $A$ $cm^3$ per second, what is the speed the water is rising when the depth of water is half of the cone's height?