Btech 1st Sem: Maths: Limit,Continuity & Differentiability

Study Materials for 1st Year B.Tech Students Paper Name: Mathematics Paper Code : M101 Teacher Name: Amalendu Singha Mahapatra Chapter – 1 Limit, Continuity and Differentiability of a Function Lecture 1. Objective: Historically, Integral Calculus developed much before the Diffential Calculus (though while teaching Differential Calculus proceeds Integral Calculus). The notions of areas/ volumes of objects attracted the attention of the Greek mathematicians like Antiphon (430 B.C.), Euclid (300 B.C.) and Archimedes (987 B.C.- 212 B.C.). Later in the 17th century, mathematicians were faced with various problems: in mechanics the problem of describing the motion of a particle mathematically; in optics the need to analyze the passage of light through a lens, which gave rise to the problem of defining tangent/ normal to a surface; in astronomy it was important to know when would a planet be at a maximum/ minimum distance from earth; and so on. The concept of limit is fundamental for the development of calculus. Calculus is built largely upon the idea of a limit and in this present chapter this idea and various related concepts will be studied in a brief manner. Limit of a Function Let x be a variable. Let x goes on taking the values 2.9, 2.99, 2.999, 2.9999, …….. Then we see x goes very near to 3 as near as we please. Mathematically we can say the quantity 3-x becomes small as small as we please. In this situation we say x tends to 3 from left. In notation x→3-. Again let x goes on taking the values 3.1, 3.01, 3.001, 3.0001, …….. Then we see x goes very near to 3 as near as we please. Mathematically we can say the quantity 3-x becomes small as small as we please. In this situation we say x tends to 3 from right. In notation x→3+. Thus we see that x→3 implies x→3+ and x→3- both. Formal Definition: Say that f (x) tends to l as x a iff given < | x - a| < , then | f (x) - l| < .

> 0, there is some

> 0 such that whenever 0

Right-handed limit We say provided we can make f(x) as close to L as we want for all x sufficiently close to a and x>a without actually letting x be a.

Left-handed limit We say provided we can make f(x) as close to L as we want for all x sufficiently close to a and x