Calculus/Limits/Exercises

• $\underset{x\to \mathrm{\infty }}{\lim }{[\cos \left(\frac{1}{x}\right)]}^{x^3\log (1+\frac{1}{x})}$
• $\underset{x\to \mathrm{\infty }}{\lim }\frac{1}{x}$ in ${ℝ}^{+}$

Answer: 0.

• $\underset{n\to \mathrm{\infty }}{\lim }\frac{{(n+1)}^{\alpha }-n^{\alpha }}{n^{\alpha -1}}$

Answer: $\alpha$

• $\underset{x\to \mathrm{\infty }}{\lim }\sqrt{x}\log (1+e^x)-x\sqrt{x}$

Answer: 0

• $\underset{x\to \mathrm{\infty }}{\lim }\sqrt{x}\log (1+e^x)-x\sqrt{x-1}$

Answer: $-\mathrm{\infty }$

• $\underset{x\to \mathrm{\infty }}{\lim }\frac{\sqrt[x]{x+\sin x}{(2+\sin x)}^x}{x!}$
• $\underset{x\to \mathrm{\infty }}{\lim }\frac{\sqrt{x^3+x}-x\sqrt{x}}{x+6\sin x}$