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Introduction to Limits | Mathematics (Maths) Class 11 - Commerce PDF Download

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 | Mathematics (Maths) Class 11 - Commerce  such that if Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

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 | Mathematics (Maths) Class 11 - Commerce

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

Introduction to Limits | Mathematics (Maths) Class 11 - Commerce= Introduction to Limits | Mathematics (Maths) Class 11 - Commerce  =    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 | Mathematics (Maths) Class 11 - Commerce

(a) Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

(b) Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

(c) Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

(d) Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

(e) Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

(f) Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

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 | Mathematics (Maths) Class 11 - Commerceand  Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

(c) Since the left and right limits are different, we conclude that Introduction to Limits | Mathematics (Maths) Class 11 - Commerce g(x) does not exist.

The graph also show that

Introduction to Limits | Mathematics (Maths) Class 11 - Commerce and  Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

(f) This time the left and right limits are the same and so, we have Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

Despite this fact, notice that Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

Ex.2 If f(x) = Introduction to Limits | Mathematics (Maths) Class 11 - Commerce determine whether Introduction to Limits | Mathematics (Maths) Class 11 - Commerce exists.

Sol.

Introduction to Limits | Mathematics (Maths) Class 11 - Commerce  

             Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

The right and left-hand limits are equal. Thus, the limit exists and   Introduction to Limits | Mathematics (Maths) Class 11 - Commerce The graph of f is shown in the figure.

Ex.3 Evaluate Introduction to Limits | Mathematics (Maths) Class 11 - Commercedenotes the greatest integer function)

Sol.

Let P = Introduction to Limits | Mathematics (Maths) Class 11 - CommerceIntroduction to Limits | Mathematics (Maths) Class 11 - Commerce

Introduction to Limits | Mathematics (Maths) Class 11 - Commerce R.H.L. = Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

∴ L.H.L. = Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

   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 | Mathematics (Maths) Class 11 - Commerce

Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

Introduction to Limits | Mathematics (Maths) Class 11 - Commerce  provided f is continuous at x = m. 

Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

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

Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

Sol. Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

  Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

= 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 | Mathematics (Maths) Class 11 - Commerce

Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

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 | Mathematics (Maths) Class 11 - Commerce  This function has no limit at any real number c.

E. INDETERMINANT  FORMS : Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

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

Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

Introduction to Limits | Mathematics (Maths) Class 11 - Commerce

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.

The document Introduction to Limits | Mathematics (Maths) Class 11 - Commerce is a part of the Commerce Course Mathematics (Maths) Class 11.
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FAQs on Introduction to Limits - Mathematics (Maths) Class 11 - Commerce

1. What is a limit in calculus?
Ans. In calculus, a limit is a fundamental concept that describes the behavior of a function as its input approaches a certain value. It is used to determine the value that a function approaches as the input gets closer and closer to a specific point, without actually reaching that point.
2. How do you find the limit of a function?
Ans. To find the limit of a function, you can evaluate the function at values of the input that are very close to the desired point. By approaching the desired point from both sides, you can observe the trend of the function's output and determine if it approaches a particular value or if it diverges.
3. What is the importance of limits in calculus?
Ans. Limits play a vital role in calculus as they provide a precise way to define and analyze important concepts such as continuity, differentiability, and integrability. They allow us to study and understand the behavior of functions at specific points or as the input approaches specific values.
4. Can a function have a limit at a point but not be defined at that point?
Ans. Yes, it is possible for a function to have a limit at a point where it is not defined. The existence of a limit only depends on the behavior of the function near the point of interest, not necessarily its value at that point. As long as the function approaches a specific value as the input approaches the desired point, the limit exists.
5. Are all limits finite?
Ans. No, not all limits are finite. Some limits can be finite, while others can be infinite. A limit can also be undefined if the function's output does not approach a specific value as the input approaches the desired point. The behavior of the function determines whether the limit is finite, infinite, or undefined.
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