Calculus/Limits/Exercises


 * $$\lim_{x \to \infty} \left [ \cos{\left ( \frac 1 x \right )} \right ]^{x^3 \log{(1 + {{1}\over{x}})}}$$
 * $$\lim_{x \to \infty} \frac{1}{x}$$ in $$\mathbb{R}^+$$
 * Answer: 0.


 * $$\lim_{n \to \infty} \frac{{\left (n+1 \right )}^\alpha - n^\alpha}{n^{\alpha-1}}$$
 * Answer: $$\alpha$$


 * $$\lim_{x \to \infty} \sqrt{x} \log {\left ( 1 + e^x \right )} - x \sqrt{x} $$
 * Answer: 0


 * $$\lim_{x \to \infty} \sqrt{x} \log {\left ( 1 + e^x \right )} - x \sqrt{x-1} $$
 * Answer: $$-\infty$$


 * $$\lim_{x \to \infty} \frac{\sqrt[x]{x+\sin{x}} \left (2+\sin{x} \right )^x}{x!}$$
 * $$\lim_{x \to \infty} \frac{\sqrt{x^3+x}-x \sqrt{x}}{x+6 \sin{x}}$$