Limits, Chapter Notes, Class 11, Maths PDF Download

Introduction to Limits - Limits and Derivatives, Class 11, Maths

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   such that if

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

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

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

(a)

(b)

(c)

(d)

(e)

(f)

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

(c) Since the left and right limits are different, we conclude that g(x) does not exist.

The graph also show that

(f) This time the left and right limits are the same and so, we have

Despite this fact, notice that

Ex.2 If f(x) =  determine whether exists.

Sol.

The right and left-hand limits are equal. Thus, the limit exists and  

The graph of f is shown in the figure.

Ex.3 Evaluate denotes the greatest integer function)

Sol.

Let P =

R.H.L. =

∴ L.H.L. =    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.

1. Sum Rule : = L + K

2. Difference Rule : = L - K

3. Product Rule :

4. Quotient Rule :     provided

5. Constant Multiplication Rule :

6. Composition Rule : provided f is continuous at x = m.

For example

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

(a)

(b)

Sol. (a)

(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.

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) = This function has no limit at any real number c.

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

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

Sol. (i) No

(ii) Yes 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) 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.

F. Methods of evaluating limits

(Rationalization, Factorization and Cancellation of Common Factors)

Ex.6 Evaluate

Sol.    =

Ex.7 Evaluate

Sol.

Ex.8 Evaluate 

Sol.

Ex.9 Evaluate  .

Sol.

Ex.10 Find

Sol.

Ex.11 Find

Sol.

Ex.12 Find

Sol.

Ex.13 Evaluate

Sol.

G. standard theorem

(1) Sandwich Theorem / Squeeze Play Theorem

Statement : If for all x in an open interval containing c, except possibly at c itself, and if

Ex.14 Find

Sol.

Ex.15 Let a function f(x) be such that

Sol.

Ex.16 Use Sandwich theorem to evaluate:

Sol.

terms of the sequence are decreasing and number of terms are (2n + 2)