Limit of a function

Limit x→a f(x) = l, means | f(x)-l| can be made as small as we like by making | x-a| sufficiently small without making x = a.

Since we undertake to make | f(x)-l| as small as required, we will be told how small | f(x)-l| is to be made. We will be given a criterion ε>0 and we must make | f(x)-l| < ε for values of x near a. We must do this by making | x-a| small enough. Hence we must find δ>0 such that when | x-a| < δ (and x≠a), | f(x)-l| will be less than ε.

| f(x)-l| < ε means f(x) belong to the ε neighbourhood of l and | x-a| < δ (and x≠a) means x belongs to the deleted δ neighbourhood of a.

Note:

The open interval (a- δ, a+ δ) whose length is 2 δ and whose midpoint is a, is called the δ-interval of a. Open interval means a- δ, and a+ δ are not part of the interval.

{x/x Є(a- δ,a+ δ), x≠0} is called the deleted δ-neighbourhood of a.

Definition of limit

Lim x→a f9x) = 1 if, given any ε>0, we can find δ>0 such that | f(x)-l| < ε whenver 0<| x-a| < δ.

Algebra of limits

1. lim x→a [f(x) ±g(x)] = lim x→a f(x) ±lim x→ag(x)

2. lim x→a [k.f(x)] = k lim x→af(x)

3. lim x→a[f(x).g(x)] = [lim x→a f(x)][ lim x→a g(x)]

4. lim x→a [f(x)/g(x)] = [lim x→a f(x)]/[ lim x→ag(x)] provided lim x→a g(x) ≠

0.

(Note: most of the problems given in exercises in the chapter are on applying this condition. Check whether the denominator becomes zero at ‘a’ and then check if the numerator also becomes zero at the ‘a’. If both become zero, it means there is a common factor and one has to remove the common factor to find the limit)

5. If f(x)

Standard limits

1. lim x→a x = a

2. lim x→a kx = ka, k Є R

3. lim x→a x

^{k}= a

^{k }, k Є R

4. lim x→a k = k

5. lim x→a (x

^{k }–a

^{k })/(x-a) = na

^{n-1}, nk Є N

6. lim x→a sin x = sin α

7. lim x→a cos x = cos α

8. lim x→a (sin x)/x = 1, where x is measured in radians

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