This document provides an overview of key concepts in calculus including functions, limits, continuity, and differentiation. It defines a function as a relationship where each input has a single output. Limits describe the behavior of a function as the input value approaches a number. A function is continuous if its limit equals the function value. A function is differentiable at a point if the limit of its difference quotient exists, with the left and right derivatives needing to be equal. Examples are provided to illustrate these fundamental calculus topics.

1. Section 2 : Calculus
Topic 1 : Functions of Single Variable Limit,
Continuity and Differentiability

2. Functions
 A function is a special relationship where each input has a single output.
 It is often written as "f(x)" where x is the input value.
 e.g. - f(x) = x/2 is a function, because each input "x" has a single output "x/2“
f(8) = 8/2 = 4, f(-24) = -12
 Explicit function : An explicit function is one which is given in terms of
the independent variable. i.e. z = f(x1, x2,….,xn)
e.g. y = x2 + 3x – 8
 Implicit functions : on the other hand, are usually given in terms of both dependent and
independent variables. i.e. Ф(z, x1, x2,….,xn) = 0 e.g. y + x2 - 3x + 8 = 0
Dependent
Variable
Independent
Variables

3. Functions
 Composite function : z = f(x, y) where, x = Ф(t) and y = ψ(t)
Some special functions
 Even function : f(-x) = f(x) e.g. – cos x, x2
 Odd function : f(-x) = - f(x)
 Modulus function : f(x) = |x|= x ; x > 0
= -x ; x < 0
= 0 ; x = 0
 Greatest integer function : For all real numbers, x, the greatest integer function returns the
largest integer less than or equal to x. f(x) = [x] = n є z |where n ≤ x < n+1.
e.g. [7.2] = 7

4. Functions
 Symmetric properties of the curve:
Let f( x, y) = c be the equation of the curve
1) If f( x, y) contains only even power of x i.e. f(-x, y) = f( x, y) then it is symmetric about y axis
2) If f( x, y) contains only even powers of y i.e. f( x, -y) = f( x, y) then it is symmetric about x
axis
3) If f( x, y) = f ( y, x) then the curve is symmetric about y = x

5. Limits
 Limit of a function: let f(x) be defined in neighbourhood of a є R, then l є R is said to be limit
f(x) as x approaches a if for given є > 0 & δ > 0 such that |f(x) – l|< є whenever |x – a|< δ.
 lim
𝑥→𝑎
𝑓(𝑥) = l
 Left limit : when x < a, lim
𝑥→𝑎−
𝑓(𝑥) = lim
ℎ→0
𝑓(𝑎 − ℎ) where h = a - x
 Right limit : when x < a, lim
𝑥→𝑎+
𝑓(𝑥) = lim
ℎ→0
𝑓(𝑎 + ℎ) where h = x – a
 Limit exists only if lim
𝑥→𝑎−
𝑓(𝑥) = lim
𝑥→𝑎+
𝑓(𝑥)
 L’ Hospital’s rule : lim
𝑥→𝑎
𝑓(𝑥)
𝑔(𝑥)
= lim
𝑥→𝑎
𝑓′(𝑥)
𝑔′(𝑥)
[ as
0
0
or
∞
∞
]

6. Limits
 Example : Applying L’ Hospitals' rule
lim
𝑥→0
1−cos 3𝑥
𝑥 sin 2𝑥
[
0
0
]
lim
𝑥→0
3 sin 3𝑥
sin 2𝑥+2𝑥 cos 2𝑥
lim
𝑥→0
9 cos 3𝑥
2 𝑐𝑜𝑠2𝑥+2 cos 2𝑥−4𝑥 sin 2𝑥
=
9
4
IMP

7. Limits
 If we have 1∞ form, lim
𝑥→𝑎
𝑓(𝑥)g(x) = 𝑒
lim
𝑥→𝑎
𝑔 𝑥 [𝑓 𝑥 −1]
 e.g. lim
𝑥→0
1 − sin 𝑥 ^ (
1
sin x
)
 𝑒
lim
𝑥→0
1−sin 𝑥 −1
sin 𝑥
 𝑒
lim
𝑥→0
−1

1
𝑒

8. Continuity
 Continuity at a point : A function is said to be continuous at a point x = a, if lim
𝑥→𝑎
𝑓(𝑥) = 𝑓(𝑎)
 Continuity in an interval : A function f(x) is said to be continuous in [a, b] if it satisfies following
three conditions
1. f(x) is continuous ∀ 𝑥 ∈ (𝑎, 𝑏)
2. lim
𝑥→𝑎+
𝑓(𝑥) = 𝑓(𝑎)
3. lim
𝑥→𝑏−
𝑓(𝑥) = 𝑓(𝑏)
e.g. f(x) = 0 ; x = 0
=
1
2
- x ; 0 < x <
1
2
=
1
2
; x =
1
2
=
3
2
- x ;
1
2
< x < 1 = 1 ; x ≥ 1

9. Continuity
e.g. f(x) = 0 ; x = 0
=
1
2
- x ; 0 < x <
1
2
=
1
2
; x =
1
2
=
3
2
- x ;
1
2
< x < 1
= 1 ; x ≥ 1 which of the following is true.
a) f(x) is right continuous at x = 0
b) f(x) is discontinuous at x =
1
2
c) f(x) is continuous at x = 1
d) b & c ans : (b)

10. Differentiation
 A function f(x) is said to be differentiable at a point x = c, if lim
𝑥→𝑐
𝑓(𝑥)−𝑓(𝑐)
𝑥 −𝑐
exists and it is
represented by 𝑓′(𝑐).
 Left Hand Derivative : lim
ℎ→0
𝑓(𝑐−ℎ)−𝑓(𝑐)
−ℎ
, h = c - x
 Right Hand Derivative : lim
ℎ→0
𝑓(𝑐+ℎ)−𝑓(𝑐)
ℎ
, h = x - c
Necessary condition for function to be differentiable is LHD = RHD
e.g. f(x) = |x| is not differentiable at x = 0
LHD = lim
ℎ→0
𝑓(0−ℎ)−𝑓(0)
−ℎ
= lim
ℎ→0
|−ℎ|−0
−ℎ
= -1
RHD = lim
ℎ→0
𝑓(0+ℎ)−𝑓(0)
ℎ
= lim
ℎ→0
|ℎ|−0
ℎ
= 1
LHD ≠ RHD  Not differentiable

11. Differentiation
 Note:
 Every differentiable function is a continuous function
 But every continuous function is not differentiable