Limits - Prof. Reem Alharbi

LIMITS

Hello Students; we will explore limits and their benefits in this lecture.  

We use limits to describe the way a function ƒ varies. Some functions vary continuously; small changes in x produce only small changes in F(x). Let ƒ(x) be defined on an open interval about except possibly at itself. If F(x) gets arbitrarily close to L (as close to L as we like) for all x sufficiently close to  we say that F approaches the limit L as x approaches and we write which is read “the limit of ƒ(x) as x approaches is L”. 

Lim F (x) = L (as x tends to x_o)

x=L^- signifies the Left Hand Limit (LHL)

x=L⁺ signifies the Right Hand Limit (RHL)

A function F(x) has a limit if both the LHL and RHL are equal. If they are not equal the function will have no limit.

Examples

F(x)=x²-5/x-2

Solution.

Lim F(x) does not exist at x=2 since the denominator will become zero at that point.

*****

Lim x-1/ x²-1 at point x=1

Solution.

We begin by factorizing the denominator. It will become x-1/(x-1)(x+1)

The x-1 in the denominator cancels with the one in the numerator. Thus it will be

1/x+1

At this point we substitute with x=1

The limit thus becomes 0.5

The Properties of Limits

Suppose we have a constant c and the Lim f(x) and lim g(x) exists. Then;

1. Sum Rule - Lim [F(x)+G(x)]= lim F(x) + lim G(x) as x tends to a
2. Difference Rule - Lim [F(x)-G(x)]= lim F(x) - lim G(x) as x tends to a
3. Constant multiple Rule - Lim [cF(x)]=c lim F(x) as x tends to a
4. Product Rule - Lim [F(x)*G(x)]= lim F(x) * lim G(x) as x tends to a
5.  Quotient Rule - Lim [F(x)/G(x)]= lim F(x) / lim G(x) as x tends to a