Calculus Notes: Limit, Continuity, and
Differentiability
Student Notes
1 Limit
1.1 Definition
A function f (x) is said to tend to limit ’l’ as x tends to a if, corresponding
to any arbitrary positive number ϵ, however small, there exists a positive
number δ, such that:
|f (x) − l| < ϵ
for all values of x for which 0 < |x − a| < δ.
We write it as:
lim f (x) = l
x→a
Right Hand Limit (R.H.L)
A function f (x) is said to tend to the limit l from the right, if corresponding
to any arbitrary positive number ϵ, however small, there exists a positive
number δ such that:
|f (x) − l| < ϵ
for all values of x for which a < x < a + δ. It is denoted by f (a + 0).
Left Hand Limit (L.H.L)
A function f (x) is said to tend to the limit l from the left, if corresponding
to any arbitrary positive number ϵ, however small, there exists a positive
number δ such that:
|f (x) − l| < ϵ
for all values of x for which a − δ < x < a. It is denoted by f (a − 0).
1
, 2 Algebra of Limits (from Page 2)
1. Existence Condition
limx→a f (x) exists only if:
lim f (x) = lim+ f (x)
x→a− x→a
2. Algebraic Operations
Let limx→a f1 (x) = l1 and limx→a f2 (x) = l2 . Then:
• Addition/Subtraction:
lim [f1 (x) ± f2 (x)] = l1 ± l2
x→a
• Multiplication:
lim [f1 (x) · f2 (x)] = l1 × l2
x→a
• Division:
f1 (x) l1
lim = (provided l2 ̸= 0)
x→a f2 (x) l2
2
Differentiability
Student Notes
1 Limit
1.1 Definition
A function f (x) is said to tend to limit ’l’ as x tends to a if, corresponding
to any arbitrary positive number ϵ, however small, there exists a positive
number δ, such that:
|f (x) − l| < ϵ
for all values of x for which 0 < |x − a| < δ.
We write it as:
lim f (x) = l
x→a
Right Hand Limit (R.H.L)
A function f (x) is said to tend to the limit l from the right, if corresponding
to any arbitrary positive number ϵ, however small, there exists a positive
number δ such that:
|f (x) − l| < ϵ
for all values of x for which a < x < a + δ. It is denoted by f (a + 0).
Left Hand Limit (L.H.L)
A function f (x) is said to tend to the limit l from the left, if corresponding
to any arbitrary positive number ϵ, however small, there exists a positive
number δ such that:
|f (x) − l| < ϵ
for all values of x for which a − δ < x < a. It is denoted by f (a − 0).
1
, 2 Algebra of Limits (from Page 2)
1. Existence Condition
limx→a f (x) exists only if:
lim f (x) = lim+ f (x)
x→a− x→a
2. Algebraic Operations
Let limx→a f1 (x) = l1 and limx→a f2 (x) = l2 . Then:
• Addition/Subtraction:
lim [f1 (x) ± f2 (x)] = l1 ± l2
x→a
• Multiplication:
lim [f1 (x) · f2 (x)] = l1 × l2
x→a
• Division:
f1 (x) l1
lim = (provided l2 ̸= 0)
x→a f2 (x) l2
2
