**MATHMAA**

## Probability Introduction :

Let us discuss the concept of **Limits** in this section.

One important aspect of the study of calculus is the analysis of how function values change as input values change. Basic to this study is the notion of a limit. Suppose a function f is given and suppose the x values get closer and closer to some number say a, then corresponding outputs f(x) get closer and closer to a number b, that number b is called the limit of f(x) as x approaches to a.

Example: Let f(x) =x+2 and select x-values that get closer and closer to 2.

Let us observe how the function value changes from left to right as x approaches to 2 from left to right.

As x approaches to 2 from either side , f(x)=x+2 approaches to 4 .

An arrow, -->, is often used to stand for the words " approaches from either side". Thus the statement above can be written as :

As \(x\rightarrow\) 2, \(x+2\rightarrow\)4.

The number 4 is said to be limit of f(x)=x+2 as x approaches to 2 from either side. we can write this statement as follows:

\(\lim_{x\rightarrow 2}f(x)=\lim_{x\rightarrow 2} (x+2) = 4\) .

Thus we can define the limit as follows:

## Definition :

As x approaches a, the limit of f(x) is L ,written as :

\(\lim_{x\rightarrow a}f(x)= L\)

When we write \(\lim_{x\rightarrow a}f(x)\), we are indicating that x is approaching a from both sides. If we want to be specific about the side from which the x-values approach the value a, we use the notation

\(\lim_{x\rightarrow a-} f(x)\) to indicate the limit from the left (x<a), and

\(\lim_{x\rightarrow a+} f(x)\) to indicate the limit from the right (x>a).

These are called left-hand and right-hand limits respectively. In order for a limit to exist, both the left-hand and right-hand limits must exist and be the same. This leads to the definition as follows:

As x approaches a, the limit of f(x) is L if the limit from the left exists and the limit from the right exists and both limits are L . That is,

if \(\lim_{x\rightarrow a-} f(x) =\lim_{x\rightarrow a+}f(x)=L\), then \(\lim_{x\rightarrow a}f(x) = L\) .

Let us do some examples on limits :

Examples:

#1 : Use the Table and find the limits of the following:

a) f(x)=2x-1 find \(\lim_{x\rightarrow 2}f(x)\) .

Sol: Let us find the table values of f(x)=2x-1 in the following table.

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