Limits & Continuity and Differentiation Study Notes

By Akhil Gupta|Updated : November 10th, 2020

1. Limits

Let us consider a function f(x) defined in an interval l. if we see the behavior of f(x) become closer and closer to a number l as x  a then l is said to be limit of f(x)at x=a.

1.1 Left Hand Limit

Let  function f(x) is said to approach l as x →a from left if for an arbitrary positive small number ε ,a small positive number δ (depends on ε) such that 

byjusexamprep

1.2 Right Hand Limit

Let  function f(x) is said to approach l as x → a from right if for an arbitrary positive small number ε ,a small positive number δ (depends on ε) such that

byjusexamprep

1.3.   L- Hospital Rule:

When 

byjusexamprep

must not be zero, where f(n) and g(n) are nth derivative of f(x) and g(x).

1.3.1. L- Hospital Rule for the form (∞ – ∞, 0 × ∞):

byjusexamprep

1.3.2. L-Hospital Rule for the form (0°, 1, ∞°):

byjusexamprep

2. CONTINUITY

A function y = f(x) is said to be continuous if the graph of the function is a continuous curve. On the other hand, if a curve is broken at some point say x = a, we say that the function is not continuous or discontinuous.

2.1.   Definition:

A function f(x) is said to be continuous at x = a if and only if the following three conditions are satisfied:

byjusexamprep

2.2.   Properties of continuous functions:

(i) A function which is continuous in a closed interval is also bounded in that interval.

(ii) A continuous function which has opposite signs at two points vanishes at least once between these points and vanishing point is called root of the function.

(iii) A continuous function f(x) in the closed interval [a, b] assumes at least once every value between f(a) and f(b), it being assumed that

 f(a) ≠ f(b).

 

3. DIFFERENTIABILITY

byjusexamprep

Note.

A Necessary condition for the Existence of a Finite Derivative

Continuity is a necessary but not the Sufficient for the existence of a finite derivatives.

 

4.   Fundamental Theorem:

4.1. Rolle’s Theorem:

If

(i)   f(x) is continuous is the closed interval [a, b],

(ii)  f’(x) exists for every value of x in the open interval (a, b) and

(iii) f (a) = f(b), then there is at least one value c of x in (a, b) such that f’ (c) = 0.

4.2    Lagrange’s Mean-Value Theorem:

If

 (i) f(x) is continuous in the closed interval [a, b], and

(ii) f’(x) exists in the open interval (a, b),

 then there is at least there is at one value c of x (a, b),

byjusexamprep

4.3.   Cauchy’s Mean-value theorem:

If (i) f(x) and g(x) be continuous in [a, b]

(ii) f’(x) and g’(x) exist in (a, b) and

(iii) g’(x) ≠ 0 for any value of x in (a, b),

byjusexamprep

limit

 

Basic Differentiation Formulas.

Suppose f  and g are differentiable functions, c is any real number, then

s

d

Thanks

The Most Comprehensive Exam Prep App.

Download BYJU'S Exam Prep, Best gate exam app for Preparation

Comments

write a comment

ESE & GATE ME

Mechanical Engg.GATEGATE MEHPCLBARC SOESEIES MEBARC ExamISRO ExamOther Exams

Follow us for latest updates