Introduction
We have discussed a function and its graph in chapter I. As a review, we give the definition of a function, its domain, range and its graph once again before defining the limit of a function.
We deal with limits and continuity which are quite fundamental for the development of calculus. These two concepts are closely linked together with the involvement of the concept of limit in the definition of continuity. So in the sequence, limit comes first and it is proper to begin with some discussion about it.
The discussion is initiated with some examples so as to give some intuitive idea about it. Then follows the precise definition of the limit. The same line of approach is being followed in the case of continuity as well. We shall also mention some limit theorems and properties of continuous functions without proof.
Function
Let X and Y be two non-empty sets. Then a function f from X to Y is a rule which assigns a unique element of Y to each element of X. The unique element of Y which ƒ assigns corresponding to an element x ∈ X is denoted by f(x).
So, we also write y = f(x) The symbol f : X→Y usually means ‘f is a function from X to Y’. The element f(x) of Y is called the image of x under the function f.
Value of the Function
If f is a function from X to Y and x = a is an element in the domain of f, then the image f(a) corresponding to x = a is said to be the value of the function at x = a.
If the value of the function f(x) at x = a denoted by f(a) is a finite number, then f(x) exists or is defined at x = a otherwise, f(x) does not exist or is not defined at x = a.
For example:
i. y = f(x) = 3x + 5 exists or is defined at x = 2as
f(2) = 3 * 2 + 5 = 11 is a finite number
ii. y = f(x) = 1/(x – 1) is not defined at x = 1 as
f(1) = 1/0 is not a finite number.
Hence f(x) does not exist or is undefined at x = 1.
Exercise – 15.1
Trigonometry Formula
We need to remember trigonometry formula that we have read in class 10. We need them while solving the problems given in this exercise. If you do remember all the formula then that’s great for you. Even if you don’t remember them, don’t you worry. I have listed all the formula needed to finish this exercise. So have a look at some of the necessary trigonometry formulas.
- Sin(A+B) = sinA.cosB + cosA.sinB
- Sin (A-B) = sinA.cosB – cosA.sinB
- Cos(A + B) = cosA.cosB – sinA.sinB
- Cos(A – B) = cosA.cosB + sinA.sinB
- sin 2A = 2sinA.cosA
- Cos2A = 2cos²A – 1 or 1- 2sin²A
- 1 – cos 2A = 2sin²A
- 1 – cosA = 2sin²A/2
- 1 + cos2A = 2cos²A
- 1 + cosA = 2Cos²A/2
- sinC + sinD = 2sin (C+D)/2×cos(C-D)/2
- sinC – sin D = 2cos (C+D)/2×sin(C-D)/2
- cosC+ cosD = 2cos (C + D)/2×cos (C-D)/2
- cosC- cosD= 2sin(C+D)/2×sin(D-C)/2
- cos²A – sin²A =cos2A
Well these are some of the formula that we already have learned in grade 10 previous year.
Limits Of Trigonometric Functions
Apart from previous year trigonometry formula, we need to remember some extra formula this year in grade 11. This formula are known as limit formula for trigonometric functions. Some of the important formula are given below;
Limits Log and Exponential Functions
We also need formula of log and exponential function while finding their limits. Some of the necessary formula to find the limits of log and exponential functions are given below;
Exercise – 15.2
Right hand limit
A function f(x) is said to have the right hand limit l₁ at x = a as x approaches a through value greater than a (i.e. x approaches a from the right) and symbolically it is written as limₓ→ₐ+ f(x) = l₁. The right hand limit of f(x) at x = a is also written as limₓ→ₐ+⁰ f(x) f(a + 0).
Left hand limit
A function f(x) is said to have the right hand limit l₂ at x = a as x approaches a through value less than a (i.e. x approaches a from the left) and symbolically it is written as limₓ→ₐ- f(x) = l₂. The left hand limit of f(x) at x = a is also written as limₓ→ₐ-⁰ f(x) f(a – 0).
Continuity of a Function
The intuitive idea of a continuous functions f in the interval [a, b] gives the impression that the graph of the function f in this interval is a smooth curve without any break in it. Actually this curve is such that it can be drawn by the continuous motion of pencil without lifting it in a sheet of paper. Similarly, a discontinuous function gives the picture consisting of disconnected curves.
