Limits And Derivatives

When the input reaches a particular value, a function or sequence hits its limit. All significant notions in calculus and mathematical analysis, such as continuity, derivatives, and integrals, are defined in terms of limits. Let's examine the meaning and illustration of function limitations as well as some guidelines, characteristics, and limit instances.

Mathematical limits are unique real numbers. Consider the limit of a real-valued function "f" and a real number "c," which is generally defined as

lim_x→c f(x)=L

It says, “The limit of f of x as x approaches c equals L.”

The "lim" denotes the limit, and the right arrow denotes the fact that function f(x) approaches the limit L as x approaches c.

Given: Let f(x) = x^m
Then, the derivative is:

We are given the condition:

Substitute the derivative expression:
Divide both sides by (assuming ):

We now solve for values of such that the equation holds for all .
Try m = 0: Then, and So, ✅ Holds true

Try m = 2: Then, LHS: RHS: ✅ Holds true

These are the only values that satisfy the condition in general.

Final Answer: (A): 0 and (C): 2