Introduction to Limits JEE Notes | EduRev

Mathematics (Maths) Class 11

JEE : Introduction to Limits JEE Notes | EduRev

The document Introduction to Limits JEE Notes | EduRev is a part of the JEE Course Mathematics (Maths) Class 11.
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A. Definition of Limit 

Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement Introduction to Limits JEE Notes | EduRev  such that if Introduction to Limits JEE Notes | EduRev

B. The Existence of a Limit 

Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L if and only if

Introduction to Limits JEE Notes | EduRev

In other words limit of a function f(x) is said to exist as, x→a when

Introduction to Limits JEE Notes | EduRev= Introduction to Limits JEE Notes | EduRev  =    A finite quantity .

Ex.1 The graph of a function g is shown in the figure. Use it to state the values (if they exist) of the following

Introduction to Limits JEE Notes | EduRev

(a) Introduction to Limits JEE Notes | EduRev

(b) Introduction to Limits JEE Notes | EduRev

(c) Introduction to Limits JEE Notes | EduRev

(d) Introduction to Limits JEE Notes | EduRev

(e) Introduction to Limits JEE Notes | EduRev

(f) Introduction to Limits JEE Notes | EduRev

Sol. From the graph we see that the values of g(x) approach 3 as x approaches 2 from the left, but they approach 1 as x approaches 2 from the right. Therefore

Introduction to Limits JEE Notes | EduRevand  Introduction to Limits JEE Notes | EduRev

(c) Since the left and right limits are different, we conclude that Introduction to Limits JEE Notes | EduRev g(x) does not exist.

The graph also show that

Introduction to Limits JEE Notes | EduRev and  Introduction to Limits JEE Notes | EduRev

(f) This time the left and right limits are the same and so, we have Introduction to Limits JEE Notes | EduRev

Despite this fact, notice that Introduction to Limits JEE Notes | EduRev

Ex.2 If f(x) = Introduction to Limits JEE Notes | EduRev determine whether Introduction to Limits JEE Notes | EduRev exists.

Sol.

Introduction to Limits JEE Notes | EduRev  

             Introduction to Limits JEE Notes | EduRev

The right and left-hand limits are equal. Thus, the limit exists and   Introduction to Limits JEE Notes | EduRev The graph of f is shown in the figure.

Ex.3 Evaluate Introduction to Limits JEE Notes | EduRevdenotes the greatest integer function)

Sol.

Let P = Introduction to Limits JEE Notes | EduRevIntroduction to Limits JEE Notes | EduRev

Introduction to Limits JEE Notes | EduRev R.H.L. = Introduction to Limits JEE Notes | EduRev

∴ L.H.L. = Introduction to Limits JEE Notes | EduRev

   Hence P = 1.

C. FUNDAMENTAL THEOREMS ON LIMITS

Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits.  Introduction to Limits JEE Notes | EduRev

Introduction to Limits JEE Notes | EduRev

Introduction to Limits JEE Notes | EduRev

Introduction to Limits JEE Notes | EduRev  provided f is continuous at x = m. 

Introduction to Limits JEE Notes | EduRev

Ex.4 Evaluate the following limits and justify each step.

Introduction to Limits JEE Notes | EduRev

Introduction to Limits JEE Notes | EduRev

Sol. Introduction to Limits JEE Notes | EduRev

  Introduction to Limits JEE Notes | EduRev

Introduction to Limits JEE Notes | EduRev

= 2(52) – 3(5) + 4 = 39

(b) We start by using laws of limit, but their use is fully justified only at the final stage when we see that the limits of the numerator and denominator exist and the limit of the denominator is not  0.

Introduction to Limits JEE Notes | EduRev

Introduction to Limits JEE Notes | EduRev

Introduction to Limits JEE Notes | EduRev

D. NON-EXISTENCE OF LIMIT

Three of the most common types of behaviour associated with the non-existence of a limit.

1. f(x) approaches a different number from the right side of c than it approaches from the left side.

2. f(x) increases or decreases without bound as x approaches c.

3. f(x) oscillates between two fixed values as x approaches c.

There are many other interesting functions that have unusual limit behaviour. An often cited one is the Dirichlet function f(x)   Introduction to Limits JEE Notes | EduRev  This function has no limit at any real number c.

E. INDETERMINANT  FORMS : Introduction to Limits JEE Notes | EduRev

Ex.5 Which of the following limits are in indeterminant forms. Also indicate the form

Introduction to Limits JEE Notes | EduRev

Introduction to Limits JEE Notes | EduRev

Introduction to Limits JEE Notes | EduRev

Introduction to Limits JEE Notes | EduRev

Sol. (i) No (ii) Yes 0/0 form (iii) Yes 0 × ∝ form (iv) Yes (∝ – ∝) form

(v) Yes, 00 form (vi) Yes ∝0 form (vii) Yes 1∝form (viii) No

Remark : 

(i)   ‘0’ doesn’t means exact zero but represent a value approaching towards zero similarly to ‘1’ and infinity.

(ii) ∝+ ∝= ∝(iii) ∝× ∝= ∝(iv) (a/) = 0 if a is finite

(v) a/0 is not defined for any a ε R.

(vi)  a b = 0, if & only if a = 0 or b = 0 and a & b are finite.

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