Limit of a Function

Students can study about Limit of a Function here. Consider the function f(x). Let the independent variable x take values near a given constant a. Then f(x) takes a corresponding set of values. Suppose that when x is close to a, the values of f(x) are close to some constant. Suppose f(x) can be made to differ arbitrarily small from A by taking values of x that are sufficiently close to a but not equal to a and that is true for all such values of x. Then f(x) is said to approach limit A as x approaches a.

If the function f(x) approaches a constant A when x approaches a in whatever manner without assuming the value a, A is said to be the limit of f(x) as x approaches a. Thus we write lim_(x->a) f(x) = A

Find the Limit of a Function

Students can learn to Find the Limit of a Function if they know what Functions are and how they behave at the given limits.

A function may approach two different limits. One where the variable approaches its limit through values larger than the limit and the other where the variable approaches its limit through values smaller than the limit. In such a case the limit is not defined but the right and left hand limit exists.

The right hand limit of a function is the value of the function approaches when the variable approaches its limit from the right. Here, we write lim_(x->a^+) f(x) = A+

The left hand limit of a function is the value of the function approaches when the variable approaches its limit from the left.

here, we write lim_(x->a^-) f(x) = A-

The limit of a function exists if and only if the left hand limit = right hand limit.

In that case, lim_(x->a^+) f(x) = lim_(x->a^-) f(x) = f(x)

Properties of limit of a function

The following are some of the properties of limits which are useful in evaluating the limit of a function.

1. lim_(x->a) k = k ( k is a constant)

2. lim_(x->a) [f(x) ± g(x)] =  lim_(x->a) f(x) ± lim_(x->a) g(x)

3. lim_(x->oo) [f(x).g(x)] = lim_(x->oo) f(x) . lim_(x->oo) g(x)

4. lim_(x->a) (f(x))/(g(x)) = (lim_(x->a)f(x))/(lim_(x->a)g(x))

5. lim_(x->a) [f(x)]n =  [ lim_(x->a) f(x)]n

Standard limit theorems:

1. lim_(x->a) (x^n- a^n)/(x - a) = nan-1

2. lim_(x->0) (e^x-1)/(x) = 1

3. lim_(x->0) (sin x)/(x) = 1

4. lim_(x->0) (1 + (1)/(n))^(n) = e



Solved Examples

Example 1: Evaluate the right hand limit of the function

f(x) = {│x – 4│/x – 4, x ≠ 4, 0 x = 4

at x = 4

Sol: (RHL of f(x) at x = 4)

= lim f(x) = lim f(4 + h) = lim │4+ h – 4│4 + h – 4

x→4+      h→0             h→0

= lim │h│/h = lim h/h = lim 1 = 1

h→0            h→0      h→0

Example 2: Let f be the function given by f(x) = x2 – a2/x – a, x ≠ a

Using (in , δ) definition show that lim f(x) = 2a

x → 0

Sol: Let in > 0 be given. In order to show that

lim f(xi) = 2a

x → a

We have to show that that for any given in > 0, there exists a number δ >0 such that

│f(x) – 2a│< in whenever 0 < │x – a│< δ

If x ≠ a, then │f(x) – 2a│= │x3 – a2/x – a│

= │(x + a) – 2a│                                         [... x ≠ a]

= │x – a│

... │f(x) – 2a│< in , if │x – a│< in

Choosing a number δ such that 0 < δ < in , we have

│f(x) – 2a│< in when whenever 0 < │x – a│< δ

Hence    lim f(x) = 2a

x → 0