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
= = 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
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.
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)
Let P =
∴ 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.
provided f is continuous at x = m.
Ex.4 Evaluate the following limits and justify each step.
= 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.
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.
E. INDETERMINANT FORMS :
Ex.5 Which of the following limits are in indeterminant forms. Also indicate the form
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
(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.