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Introduction of Limits

notes

The function f(x) = `x^2`. Observe that as x takes values very close to 0, the value of f(x) also moves towards 0 .
 `lim_(x->0)` f(x) = 0 
(to be read as limit of f (x) as x tends to zero equals zero). The limit of f (x) as x tends to zero is to be thought of as the value f (x) should assume at x = 0.
In general as x → a, f (x) → l, then l is called limit of the function f (x) which is symbolically written as `lim_(x-> a)` f(x) = l .The value of h(x) for values of x very near to 2 (but not at 2). This is somewhat strengthened by considering the graph of the function y = h(x) given here in following fig.

In all these illustrations the value which the function should assume at a given point x = a did not really depend on how is x tending to a. 
two ways x could approach a number a  either from left or from right, i.e., all the values of x near a could be less than a or could be greater than a. This naturally leads to two limits – the right hand limit and the left hand limit. Right hand limit of a function f(x) is that value of f(x) which is dictated by the values of f(x) when x tends to a from the right. 
\[ f(x) = \begin{cases} 1, & \quad x≤0\\ 2, & \quad x>0 \end{cases} \]
         
Graph of this function is shown in the following fig. 

It is clear that the value of f at 0 dictated by values of f(x) with x ≤ 0 equals 1, i.e., the left hand limit of f (x) at 0 is
`lim_(x -> 0)` f(x) = 1.
Similarly, the value of f at 0 dictated by values of f (x) with x > 0 equals 2, i.e., the right hand limit of f (x) at 0 is
`lim _(x -> 0^+)` f(x) = 2.
In this case the right and left hand limits are different, and hence we say that the limit of f (x) as x tends to zero does not exist (even though the function is defined at 0).

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