Introduction

The concept of a limit plays a central role in calculus. For example, continuity, derivative and integral require this concept. In this assignment, you are required to explore the concept of a limit.

Sample Question

Question 1

(a) Draw an equilateral triangle inscribed in a circle of radius  cm. Express the area of the triangle in term of .

(b) Repeat 1(a) with square, regular pentagon, regular hexagon, …, regular polygon with n sides in the circle of radius cm.

(c) Determine the value of correct to three decimal places when is large.

Question 2

A function    is defined by  .

(a)(i) Tabulate the values and when x equals to 0.1, 0.01, 0.001, …. Deduce value of .

(a)(ii) Tabulate the values and  for , 3 and 4, when equals to 0.1, 0.01, 0.001, ….  Deduce value of  .

(b) Show that .

Question 3

A function g is defined by , where .

(a) Suppose is an integer, estimate using the trapezium rule with

(i) strips,

(ii) strips,

(iii) strips.

In each case, express in terms of  and determine $\lim_{t\to\infty} g(t)$.

(b) Suppose is a real number, find using integration and determine $\lim_{t\to\infty} g(t)$.

(c) Comment on your findings.