# Limits of a Function

A function *y* = *f *(*x*) is said to tend to limit *u *as *x *tends to *a*, if the difference between the values of *f* (*x*) and *u *becomes smaller and smaller, as the difference between the values of *x *and *a *reduces. It is represented as

**Example:**Consider a function

*f*(

*x*) = 5x

*x*= 2 â‡’

*f*(

*x*) = 5 x 2 = 10

*x*approaches the value 2,

*f*(

*x*) approaches the value 10.

*f*(

*x*) and

*f*(2) becomes smaller and smaller, when the difference between the values of

*x*and 2 becomes smaller.

*i.e.*, the value of

*f*(

*x*) approaches 10 as the value of

*x*approaches 2. This can be shown as

*x*approaching 2,

*i.e.*,

*x*approaching 2 from a value lesser than 2 and

*x*approaching 2 from a value greater than 2.

*x*approaches the value from a value lesser than it, is known as left hand limit. For the above table, the left hand limit is shown as

*x*approaches the value from a value greater than it, is known as right hand limit. For the above table, right hand limit is shown as

# Rules on limits

Let where*u*and

*v*are finite quantities:

# Methods of calculating limits

*Direct substitution method*

Example

Solution

Substituting

*x*= 2 in the given function, we getExample

Solution

Substituting

*x*= 3 in the given function, we get

**Note:** Any number of the form is known as an indeterminant form.

After substituting the value for the variable, if the final answer you get is an indeterminate form, then you can solve limits by the following methods:

# Factorisation method

In this method, we factorize the functions under the limit into simpler functions and see if any terms can be cancelled. Then, we proceed with substituting the value for the variable and calculating the limit.Example

Solution

Let us try solving this using direct substitution method.

Substituting

is not defined. Hence, we cannot solve the above problem by direct substitution.

Let us try factorizing the numerator and the denominator.

We get

Substituting

Substituting

*x*= 1, we getis not defined. Hence, we cannot solve the above problem by direct substitution.

Let us try factorizing the numerator and the denominator.

We get

Substituting

*x*= 1, we get

Example

Solution

Let us try solving this using direct substitution method.

Substituting

is not defined. Hence, we cannot solve the above problem by direct substitution.

Let us try factorizing the numerator and the denominator.

We get

Substituting

Substituting

*x*= 2, we getis not defined. Hence, we cannot solve the above problem by direct substitution.

Let us try factorizing the numerator and the denominator.

We get

Substituting

*x*= 2, we get# Solving by rationalization

We use this method of solving limits, when polynomials under the square-root sign are added or subtracted. First, we rationalize the function and then compute the limits by any of the methods given above.Example

Solution

If we try solving the above limit by direct substitution method, we will end up with an indeterminate form. Also, factorizing the functions will not help us as no terms would get cancelled. So, let us try to rationalize the function and then calculate its limit.

Multiplying the numerator and denominator of the function by
We get,
Now, substituting x = 0, we get

Multiplying the numerator and denominator of the function by

# Lâ€™Hospital rule

Lâ€™Hospital rule uses derivatives to help evaluate limits involving indeterminate forms. is an indeterminate form, then using Lâ€™Hospitalsâ€™ rule, where*f*â€²(

*x*) and

*g*â€²(

*x*) are the derivatives of

*f*(

*x*) and

*g*(

*x*).

Example

Solution

If we try solving the above limit by direct substitution, we will get an indeterminate form. Using Lâ€™Hospital rule, we can solve by differentiating both numerator and denominator.
Substituting

which is again an indeterminate form. Applying Lâ€™Hospital rule again, we get
Substituting
which is again an indeterminate form. Applying Lâ€™Hospital rule again, we get
Substituting
Hence,

*x*= 1, we getwhich is again an indeterminate form. Applying Lâ€™Hospital rule again, we get

*x*= 1, we get*x*= 1, we get

**Note:** As we saw in the Example 9.9, we need to go on applying Lâ€™Hospital rule until the end result is not an indeterminate form.

# Some important standard limits

Example

Solution

Given

Example

Solution

If we substitute

*x*= 0 in the given limit, we get the indeterminate form 1^{âˆž}Example

Solution

Multiplying and dividing the function by 2, we get,

Example

Solution

This is of the form which is equal to

Example

Solution

Multiplying and dividing the function within the limits by 2, we get,

Example

Solution

Let us try to convert the given limit into one of the standard forms.

Example

Solution

Let us try to convert the given limit into one of the standard forms.

Example

Solution

Let us try to convert the given limit into one of the standard forms.

**Note:**

Example

Solution

Now this is a standard limit and we know that

Example

Solution

Applying L Hospitalâ€™s rule