Problem

PART 1: Fill in the blanks.

• Only answers will be graded.

• Each answer must be clearly labeled on the answer sheet.

• 4 points are assigned to each blank.

1. (a) $\displaystyle\lim_{x\to\infty}\left(1+\frac{3}{x}\right)^{[x]} = \underline{\quad(1)\quad}$, where $[x]$ is the greatest integer which is less than or equal to $x$.

(b) $\displaystyle\lim_{x\to 0}(\cos ax)^{\frac{1}{1-\cos bx}} = \underline{\quad(2)\quad}$, where $a,b$ are constants and $ab\ne 0$.

(c) $\displaystyle\lim_{n\to\infty}\sum_{i=1}^n \frac{1}{2n\sqrt{1-\left(\frac{i}{2n}\right)^2}} = \underline{\quad(3)\quad}$.