Introduction to Limits

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

means that for every ε > 0 there exists a δ > 0 such that

In words: as x approaches c, the values f(x) can be made arbitrarily close to L by taking x sufficiently close to c (but not equal to c if f is not defined there). This is the precise ε-δ definition of limit.

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

Equivalently, the limit exists and equals a finite number when the left-hand limit and the right-hand limit at that point both exist and are equal:

=

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.

Since the left and right limits are different at x = 2, the two-sided limit does not exist there.

Therefore

The graph also shows the following one-sided limits and function values:

For part (f) the left and right limits are the same, so the two-sided limit exists:

Despite this, notice that the actual function value at that point may be different from the limit:

Ex.2 If f(x) =

determine whether

exists.

Sol.

The right-hand 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.

The standard limit laws follow (sum, difference, constant multiple, product, quotient, and composition) provided the necessary conditions (existence of limits and non-zero denominator where required) are satisfied. In particular, if f is continuous at x = m, then

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

Sol.

= 2(5²) - 3(5) + 4 = 39

(b) We start by using laws of limits, 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 are:

• Different one-sided limits: f(x) approaches different numbers from the right side of c and from the left side of c.
• Unbounded behaviour: f(x) increases or decreases without bound as x approaches c.
• Oscillation: f(x) oscillates between two or more values as x approaches c.

There are many other functions that have unusual limit behaviour. An often-cited example is the Dirichlet function, defined by

This function has no limit at any real number c, because in every interval around any point it takes both the values 0 and 1 (or other differing values), so the function does not approach any single value.

E. Indeterminate Forms

Expressions that appear to be limits of the form 0/0, ∞/∞, 0·∞, ∞-∞, 0⁰, ∞⁰, 1^∞, etc., are called indeterminate forms. Such forms do not by themselves determine the limit and further analysis is required.

Ex.5 Which of the following limits are in indeterminate 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

Remark :

• '0' does not mean the exact value zero here but denotes a quantity that approaches zero. Similarly, '1' and '∞' denote limiting tendencies.
• ∞ + ∞ = ∞.
• ∞ × ∞ = ∞.
• (a/∞) = 0 if a is finite.
• a/0 is not defined for any finite real a.
• ab = 0 if and only if a = 0 or b = 0, when a and b are finite.