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: $-\infty$

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