Suppose we have a function ‘f’ that is continuous at x = p if and only if it follows the given condition.
= lim x→pf (x) = f (p);
Condition 1: The given function f (p) is defined or we can say ‘p’ is the Domain of function f.
Condition 2: The limit
= lim x→pf (x) = M exist in the function.
Condition 3: The two number f (p) and M must be equal.
If any function follows all the three conditions then the function is continuous.
Now, we will see the composition of continuous functions.
As we know that the composition of continuous functions is continuous.
Let A, B and C are Subsets of R, let the function ‘f’ be a continuous function from A to B and let the function ‘g’ be a continuous function from B to C. Then the composition of function ‘g’ and ‘f’ is continuous. Now, we will see how to find the limit of continuous function. The general form of limit of a function is given by:
lim f [g (p)] = f [lim g (p)];
Example: Let’s take a functions f (s) = lims⇢∞ cos (1 / s); then solve the limit of the given function?
Solution: Given, f (s) = lims⇢∞ cos (1 / s);
let ‘f’ and ‘g’ in the given function.
⇨ f (s) = cos s;
⇨ g (s) = 1 / s;
In general limit can be written as:
lim f [g (s)] = f [lim g (s)];
⇨ f (s) = cos (lims⇢∞ (1 / s));
Now put the limit in the given function:
⇨ f (s) = cos (1 / ∞);
⇨ f (s) = cos (0);
As we know that the value of cos (0) is 1;
f (s) = 1;
This is how we can find the limits Of composition of continuous functions.