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

Answer: 0

Answer:

-\infty

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

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