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

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

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