The document is a lecture outline on the concepts of limits and continuity in mathematics, including definitions, examples, and applications. It discusses how limits describe a function's behavior near a point and the conditions for a function to be continuous. Furthermore, it lists various real-life applications of limits and continuity in fields like engineering.

1. DEPARTMENT OF MATHEMATICS, CGU
MM
MATHEMATICS

2. MATHEMATICS
LECTURE
Limit and continuity
DEPARTMENT OF MATHEMATICS,
CVRCE
REFERENCE BOOK: MATHEMATICS FOR CLASS XI
BY R.D. SHARMA

3. Outline
 The intuitive idea of limit
 Definition of limit
 Limits of polynomials and rational functions
 Limits of trigonometric, exponential and
logarithmic functions
 Continuity of a function at a point
 Applications

4. Introduction
Limits describe how a function behaves near a
point, instead of at that point. This simple yet
powerful idea is the basis of all of calculus.

5. Definition: Limit
A function is said to have a limit as
approaches ‘’ if, given any positive number
ε, there is a positive number δ such that for
all ,
We write,

6. Left Hand Limit (LHL):
Let be a given function then left hand limit
of at ‘’ is
Right Hand Limit (RHL):
Let be a given function then right hand limit
of at ‘’ is
Existence of a limit at a point:
= =

7. Example 1. Show that does not exist.
Solution:
LHL
RHL
LHL RHL
Thus, does not exist.

8. Example 2. Evaluate
Solution:
It is of the form .
=
=
=
=-11.

9. Example 3. Evaluate
Solution:
It is of the form .
=
=
=
=
=3.

10. Example 4. Evaluate
Solution:
=
=
=

11. Example 5. Evaluate
Solution:
=
=
=1
=3

12. Example 6. Evaluate
Solution:
It is of the form
Dividing the numerator and the denominator by

13. Example 7. Evaluate
Solution:
It is of the form
Dividing the numerator and the denominator
by we get

14. Example 8. Evaluate
Solution:
]

15. Example 9. Evaluate
Solution:
It is of the form
Dividing the numerator and the
denominator by we get

16. =2
Example 10. Evaluate
Solution:
=
=
=
=1+1=2

17. Example 11. Evaluate .
Solution:
= )
= )
= )
=
=

18. Example 12. Evaluate .
Solution:
It is of the form

19. Example 13. Evaluate .
Solution:

20. Example 14. Evaluate .
Solution:
Rationalising the numerator
]
]

21. Assignments:
Evaluate the following limits
1.

22. Definition: A function is continuous at a point if for every small there
exists such that for all
implies
continuous
function
non-continuous
function

23. Note: A function is continuous at a point if
we can draw its graph at that point without
lifting our pencil.
Arithmetical Definition: Continuity
A function is said to be continuous at if
LHL=RHL=value of the function at ‘’

24. Example 1. Examine the continuity of the
function
at
Solution:
Here,
LHL
RHL
LHL = RHL =
continuous at

25. Example 2. Examine the continuity of the
function
at
Solution:
Here,
LHL
RHL
LHL = RHL =
continuous at

26. Assignments:
1. Check the continuity of the function
at
2. Check the continuity of the function
at

27. Application
 Limits can be applied in real life. Everything has a
limit in this world we live in, some of us are not
just aware of it. Some examples of real life
application of limits are: speed limit, vehicle
capacity, limit of food we intake, limit of using the
internet, limit of medicine amount we intake etc.
 Continuity is one of the principles used by civil
engineers to develop bridges and buildings that
can tolerate different kinds of forces and climatic
conditions.

28. Thank you!