Limit, in the present context, means a constant value to which the function of a variable, say f(x) approaches as the variable, x approaches a given value say a. Thus the limit of a function, say L, is that value to which f(x) approaches as the variable x approaches a given value, a. the function, f(x) approaches the fixed constant, L in such a manner that the difference between the function, and the constant i.e. [f(x) – L], can be made smaller and smaller than any smallest possible positive number, say (read as epsilon ) as we may choose, and this difference continues to remain less than the chosen positive number ε, so long as the variable, x continues to appear nearer to a, the particular value chosen for it. This means that as the variable, x approaches closer and closer to a, the function of x f(x) approaches closer and closer to L, so that δ (the difference between f(x) and L) continues to remain less than ε, the chosen positive number, however, small i.e. δ < ε when x → a and f(x) → L.
“L is said to the limit of the function f(x) as x approaches a, if the difference between L and f(x) can be made as small as we can by taking x sufficiently nearer to a and is denoted symbolically as lim f(x) = L
x → a
The essential characteristics of the limit of a function may be outlined as under:
i. The variable x may assume any value sufficiently nearer to a on both of its lower and upper sides but never equals to a.
ii. The quantity lim f(x) depends only on the values of f(x) for x near a but not for ‘x’ equal to a.
iii. As x gets closer and closer to n the value of f(x) might not approach any fixed, number and in such a case, we may say that lim f(x) does not x a
iv. The difference between the f(x) and the limit always remains less than the smallest possible positive number as may be chosen.