(Solved):   KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY INSTITUTE OF DISTANCE LEARNING FACULTY ...

 

KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY INSTITUTE OF DISTANCE LEARNING FACULTY OF PHYSICAL AND COMPUTATIONAL SCIENCES DEPARTMENT OF MATHEMATICS B.Sc. Electrical / Mechanical Engineering Pre Bridging Math 152 Calculus Assignment (Answer all questions) 1. Define a deleted \( \frac{1}{2}- \) neighbourhood of 2 . 2. How many cluster points does the interval \( (-1,2) \) have? 3. Given that \( A=\left\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4} \ldots\right\} \), find the (a) lower bound of \( A \). (b) greatest lower bound of \( A \). (c) upper bound of \( \mathrm{A} \). (d) least upper bound of \( \mathrm{A} \). 5. Find the cluster point(s) of \( 1, \frac{1}{2}, 1, \frac{1}{3}, 1, \frac{1}{4}, 1, \frac{1}{5} \ldots \) 6. If \( T=\left\{\frac{1}{5^{n}}: n=0,1,2,3 \ldots\right\} \), then \( T \) is closed? A. True B. False 7. Find the limit point(s) of \( a_{n}=1+(-1)^{n}+\frac{1}{n} \) 8. The set \( T=\{1,2,3,4,5\} \) has no cluster point. A. True B. False 9. Let \( U_{n}=\frac{n+3}{2 n+1} \) (a) Prove that the sequence \( \left\{U_{n}\right\} \) is i. bounded ii. monotonic decreasing. 10. Evaluate the following: (a) \( \lim _{n \rightarrow \infty}\left(\frac{5 n^{2}+3 n+3}{2-n+7 n^{2}}\right) \) (b) \( \lim _{n \rightarrow \infty}\left(\frac{1}{n^{2}}\left(\frac{n^{2}-1}{n^{2}}\right)\right) \) (c) \( \left.\lim _{n \rightarrow \infty}\left(5 \sqrt{\left(1+\frac{1}{n^{2}}\right.}\right)\right) \) (d) \( \lim _{n \rightarrow \infty}\left(\frac{\sqrt{3 n^{2}-4 n+5}}{n+1}\right) \)