# Mathematics | Limits, Continuity and Differentiability

### 1. Limits –

For a function the limit of the function at a point is the value the function achieves at a point which is very close to .

Formally,
Let be a function defined over some interval containing , except that it
may not be defined at that point.
We say that, if there is a number for every number such that
whenever

The concept of limit is explained graphically in the following image –

As is clear from the above figure, the limit can be approached from either sides of the number line i.e. the limit can be defined in terms of a number less that or in terms of a number greater than . Using this criteria there are two types of limits –
Left Hand Limit – If the limit is defined in terms of a number which is less than then the limit is said to be the left hand limit. It is denoted as which is equivalent to where and .
Right Hand Limit – If the limit is defined in terms of a number which is greater than then the limit is said to be the right hand limit. It is denoted as which is equivalent to where and .

Existence of Limit – The limit of a function at exists only when its left hand limit and right hand limit exist and are equal i.e.

Some Common Limits –

L’Hospital Rule –
If the given limit is of the form or i.e. both and are 0 or both and are , then the limit can be solved by L’Hospital Rule.
If the limit is of the form described above, then the L’Hospital Rule says that –

where and obtained by differentiating and .
If after differentitating, the form still exists, then the rule can be applied continuously until the form is changed.

• Example 1 – Evaluate
• Solution – The limit is of the form , Using L’Hospital Rule and differentiating numerator and denominator

• Example 2 – Evaluate
• Solution – On multiplying and dividing by and re-writing the limit we get –

• ### 2. Continuity –

A function is said to be continuous over a range if it’s graph is a single unbroken curve.
Formally,
A real valued function is said to be continuous at a point in the domain if –
exists and is equal to .
If a function is continuous at then-

Functions that are not continuous are said to be discontinuous.

• Example 1 – For what value of is the function defined by

continuous at ?

• Solution – For the function to be continuous the left hand limit, right hand limit and the value of the function at that point must be equal.
Value of function at

Right hand limit-

RHL equals value of function at 0-

• Example 2 – Find all points of discontinuity of the function defined by –
.
• Solution – The possible points of discontinuity are since the sign of the modulus changes at these points.
For continuity at ,
LHL-

RHL

Value of at ,

Since LHL = RHL = , the function is continuous at

For continuity at ,
LHL-

RHL

Value of at ,

Since LHL = RHL = , the function is continuous at
So, there is no point of discontinuity.

### 3. Differentiability –

The derivative of a real valued function wrt is the function and is defined as –

A function is said to be differentiable if the derivative of the function exists at all points of its domain. For checking the differentiability of a function at point ,
must exist.

If a function is differentiable at a point, then it is also continuous at that point.
Note – If a function is continuous at a point does not imply that the function is also differentiable at that point. For example, is continuous at but it is not differentiable at that point.

GATE CS Corner Questions

Practicing the following questions will help you test your knowledge. All questions have been asked in GATE in previous years or in GATE Mock Tests. It is highly recommended that you practice them.

References-

Continuity – Wikipedia
Limits – Wikipedia
Differentiability – Wikipedia
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