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MATHMAA

Limits

Limits :

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.