Limits

Study Material:

Definition:

Let be a real valued function and be a real number.

Then can made to be as closed to as desired by making sufficiently closed to , then is said to be the limit of the function

And is written as, .

Example:

Let

Here is undefined but near to, like at,,

, . That is the value of approach to as approach to . So here limit of is 2.

Note:

There are essentially two way to approach could approach a number, either from left side of from right side.

Left hand limit:

The expected value of at when approaches to the left side of is said to be the left hand limit of at an is denoted

By

Right hand limit:

The expected value of at when approaches to the right side of is said to be the right hand limit of at an is denoted by .

Example:

Let

Here left hand limit

And right hand limit

But

Note:

If the left hand limit and right hand limit are equal and equal to , i.e., , then .

Algebra of limits:

Let and be two real valued function such that and both exist. Then

1. Limit of sum of two functions and is the sum of limits of these two functions, i.e.,

2. Limit of difference of two functions and is the difference of limits of these two functions, i.e.

3. Limit of product of two functions and is the product of limits of these two functions, i.e.

4. Limit of quotient of two functions and is the quotient of limits of these two functions (when the denominator is not zero), i.e.,

Note:

In particular case of (3), when, where is real number, then we have

Question. Let. Find the limit of this function at.

Solution:

.

Question. Consider a function. Find

Solution:

Question. Consider a function. Find.

Solution:

Question. Find, where is given by,

Solution:

And value of the function at is.

Limits of polynomial and rational functions:

Polynomial function:

If , where are real numbers and for some , then is said to be the polynomial function of of degree .

Now (1)

So,

(Using (1))

Rational function:

A function is said to be rational function if, where and both are polynomials and.

And here,

(1)

Now if, then limit of at exist.

And if, then two cases may arise.

Case I:

If, then the limit does not exist.

Case II:

If

Let, where is the maximum power of in and,

Where is the maximum power of in

If,

then

=

.

If , then the limit does not defined.

Theorem: For any positive integer,

Proof:

We know that (1)

(Using (1))

( Times)

Question. Evaluate the following limits.

1. 2. 3. 4.

5. 6. 7. 8.

Solution:

1.

2.

3.

=

4.

5.

6.

(Using the theorem)

7.

.

Limits of trigonometric functions:

Let us consider two real functions and with the same domain such that in the domain of definition

and for some , exist. The.

Sandwich theorem:

Let be three real functions with the same domain such that in the domain of definition such that. If for some and also then.

Example:

Evaluate the following

1. and 2.

Solution:

1. We know that for

Now and

So by sandwich theorem,

2.

Question. Evaluate

1. 2.

3. 4.

5. 6.

Solution:

1.

2.

.

3.

Put then

4.

5.

6.

()

Question. Find, where

Here the function can be expand as

And

.

Question. Let . Find the value of and when

.

Since

i.e.,

(1)

And (2)

Adding (1) and (2),

Putting the value of in (1),

.