Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE

GATE: Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE

The document Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE is a part of the GATE Course Engineering Mathematics.
All you need of GATE at this link: GATE

Limits

For a function f(x) the limit of the function at a point x = a is the value the function achieves at a point which is very close to x = a.
Formally,
Let f(x) be a function defined over some interval containing x = a, except that it may not be defined at that point.
We say that, Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE if there is a number δ for every number ϵ such that Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE whenever Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE
The concept of limit is explained graphically in the following image
Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE

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 a or in terms of a number greater than a.
Using this criteria there are two types of limits:

  1. Left Hand Limit: If the limit is defined in terms of a number which is less than a then the limit is said to be the left hand limit. It is denoted as Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE which is equivalent to x = a - h where h > 0 and h → 0.
  2. Right Hand Limit: If the limit is defined in terms of a number which is greater than a then the limit is said to be the right hand limit. It is denoted as x → a+ which is equivalent to x = a + h where h > 0 and h → 0.

Existence of Limit: The limit of a function f(x) at x = a exists only when its left hand limit and right hand limit exist and are equal and have a finite value i.e.
Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE

Some Common Limits:
Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE


L’Hospital Rule

If the given limit Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE is of the form 0/0 or ∞/∞ i.e. both f(x) and g(x) are 0 or both f(x) and g(x) 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 –
Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE
where f'(x) and g'(x) obtained by differentiating f(x) and g(x).
If after differentitating, the form still exists, then the rule can be applied continuously until the form is changed.

Example 1: Evaluate Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE

Solution: The limit is of the form 0/0, Using L’Hospital Rule and differentiating numerator and denominator
Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE
Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE
Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE
Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE
Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE

Example 2: Evaluate Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE
Solution: On multiplying and dividing by kx and re-writing the limit we get –Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE
Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE
Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE
Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE

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 f(x) is said to be continuous at a pointLimits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE in the domain if –Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE f(x) exists and is equal to Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE.
If a function f(x) is continuous at Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE then-
Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE
Functions that are not continuous are said to be discontinuous 

Example: For what value of λ is the function defined by
Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE
continuous at x = 0?

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 x=0

Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE

Right hand limit-

= Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE

= 1

RHL equals value of function at 0-

-2λ = 1

λ = -1 / 2

Differentiability

The derivative of a real valued function f(x) wrt x is the function f'(x) and is defined as –Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE
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 x = c, Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATEmust 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, f(x) = |x| is continuous at x = 0 but it is not differentiable at that point.

The document Limits, Continuity and Differentiability Notes | Study Engineering Mathematics - GATE is a part of the GATE Course Engineering Mathematics.
All you need of GATE at this link: GATE

Download free EduRev App

Track your progress, build streaks, highlight & save important lessons and more!

Related Searches

Exam

,

Objective type Questions

,

study material

,

video lectures

,

Extra Questions

,

Sample Paper

,

practice quizzes

,

Limits

,

Free

,

Continuity and Differentiability Notes | Study Engineering Mathematics - GATE

,

Continuity and Differentiability Notes | Study Engineering Mathematics - GATE

,

Semester Notes

,

Previous Year Questions with Solutions

,

shortcuts and tricks

,

Continuity and Differentiability Notes | Study Engineering Mathematics - GATE

,

MCQs

,

Important questions

,

Viva Questions

,

mock tests for examination

,

Limits

,

pdf

,

Summary

,

past year papers

,

Limits

,

ppt

;