Mathematics | Limits, Continuity and Differentiability

1. Limits –

Some Common Limits –

• 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. 1. GATE CS 2013, Question 22 2. GATE CS 2010, Question 5 3. GATE CS 2008, Question 1 4. GATE CS 2007, Question 1 5. GATE CS 2015 Set-1, Question 14 6. GATE CS 2015 Set-3, Question 19 7. GATE CS 2016 Set-1, Question 13 8. GATE CS 1998, Question 4

References-

Continuity – Wikipedia Limits – Wikipedia Differentiability – Wikipedia