The document discusses the concepts of limits, continuity, and differentiability in mathematics. It defines limits, explains the conditions under which limits exist, and outlines continuity with related examples. The document also presents theorems related to continuous functions and differentiability, emphasizing that while differentiable functions are continuous, continuous functions are not necessarily differentiable.

1. Limit Continuity and Derivability 1

2. Limit Continuity and Derivability 2
limit
A point or level beyond which something
does not or may not extend or pass.
In the dictionary of Mathematics , Limit value
is an approximate value which is very closed
and nearest value at a point. It is Written as:

3. Limit Continuity and Derivability 3
Let us consider a real-valued function “f” and the real number “c”, the limit is
normally defined as
Lim f(x)= L
x→c
In limits, we have in determinant forms such as
00,∞∞,0×∞,∞×∞,1∞,0 ,∞
∘ ∘
In these cases, we try to simplify the problem into a valid function.

4. Limit Continuity and Derivability 4
# When the limit exist?
The limit exists when: -
• there should be only one value that the
function approaches.
• The limit must be a finite value
• If the function has both limits defined at a
particular x value, and those values match,
then the limit will exist. If the values of the
one-sided limits do not match, then the
limit will not exist.

5. Limit Continuity and Derivability 5

6. Limit Continuity and Derivability 6

7. Limit Continuity and Derivability 7
• Examples

8. Limit Continuity and Derivability 8
Continuity
• The property of continuity is exhibited by various aspects of nature.
The water flow in the rivers is continuous. The flow of time in
human life is continuous i.e. you are getting older continuously. And
so on. Similarly, in mathematics, we have the notion of the
continuity of a function.
• What it simply means is that a function is said to be continuous if
you can sketch its curve on a graph without lifting your pen even
once (provided that you can draw good). It is a very straightforward
and close to accurate definition actually. But for the sake of higher
mathematics, we must define it in a more precise way. That’s what
we are going to do in this section. So let’s jump into it!

9. Limit Continuity and Derivability 9
Explain the conditions of continuity?
A function f(x) is said to be continuous at a point x =
a, in its domain( All values of ‘x’ that can be used in
a function and for which the function is continuous.)
if the following three conditions are satisfied:
• f(a) exists (i.e. the value of f(a) is finite)
• Limx→a f(x) exists (i.e. the right-hand limit = left-
hand limit, and both are finite)
• Limx→a f(x) = f(a)

10. Limit Continuity and Derivability 10

11. Limit Continuity and Derivability 11

12. Limit Continuity and Derivability 12
• Example 1: Check the continuity of the function f(x) = 3x - 7 at x
= 7.
-Solution:
Given that f(x) = 3x - 7
and x = 7 = a.
• We will find the limₓ → ₐ f(x) and f(a).
=limₓ → ₐ f(x)
= limₓ → ₇ (3x - 7)
= 3(7) - 7
= 21 – 7
= 14.
f(a) = f(7) = 3(7) - 7 = 21 - 7 = 14.
Therefore, limₓ → ₐ f(x) = f(a). Thus, f(x) is coninuous at x = 7.

13. Limit Continuity and Derivability 13
• Example 2: Prove that the following function is NOT continuous
at x = 2 and verify the same using its graph
f(x) = {x−3, if x≤2
{8, if x>2
Solution:
It is given that a = 2.
Now we will compute the limits.
LHL=limₓ → ₂₋ f(x)
= limₓ → ₂ (x - 3) = 2 - 3 = -1
Also,
RHL=limₓ → ₂₊ f(x)
= limₓ → ₂ 8 = 8
Here,
limₓ → ₂₋ f(x) ≠ limₓ → ₂₊ f(x).
• Thus, limₓ → ₂ f(x) does NOT exist and hence f(x) is NOT
continuous at x = 2.

14. Limit Continuity and Derivability 14
Differentiability
A function f(x) is said to be differentiable at the point x =
a if the derivative f’(a) exists at every point in its domain.
Theorem 1: Algebra of continuous functions:
If the two real functions, say f and g, are continuous at a
real number c, then
(i) f + g is continuous at x=c.
(ii) f – g is continuous at x=c.
(iii) f. g is continuous at x=c.
(iv)f/g is continuous at x=c, (provided g(c) ≠ 0).

15. Limit Continuity and Derivability 15
• Theorem 2: Suppose f and g are real-valued functions such that (f o
g) is defined at c. If g is continuous at c and if f is continuous at g (c),
then (fog) is continuous at c.
• Theorem 3: If a function f is differentiable at a point c, then it is also
continuous at that point.
• Theorem 4 (Chain Rule): Let f be a real-valued function which is a
composite of two functions u and v; i.e., f = v o u.
• Suppose t = u(x) and if both dt/dx and dv/dt exist, we have df/dx =
(dv/dt). (dt/dx)
• Theorem 5:
(1) The derivative of ex
with respect to x is ex
; i.e., d/dx(ex
) = ex
.
(2) The derivative of log x with respect to x is 1/x.
i.e., d/dx(log x) =1/x.
• Theorem 6 (Rolle’s Theorem): Let f : [a, b] → R be continuous on [a,
b] and differentiable on (a, b), such that f(a) = f(b), where a and b
are some real numbers. Then there exists some c in (a, b) such that
f'(c) = 0.

16. Limit Continuity and Derivability 16

17. Limit Continuity and Derivability 17

18. Limit Continuity and Derivability 18

19. Limit Continuity and Derivability 19

20. Limit Continuity and Derivability 20
• Is a point function continuous?
• Is a continuous function always differentiable?

21. Limit Continuity and Derivability 21
• Is a point function continuous?
A point function is not continuous according to
the definition of continuous function.
• Is a continuous function always differentiable?
A continuous function need not be differentiable.
If a function f(x) is differentiable at point a, then it
is continuous at point a. But the converse is not
true.

22. Limit Continuity and Derivability 22
Your Feedback is Most Important in correction in
my next presentation.
(तपाईंको प्रतिक्रिया मेरो अर्को प्रस्तुतीकरणमा सुधारको
लागि सबैभन्दा महत्त्वपूर्ण छ।)