mathman said:
MathewsMD said:
I presume you mean "let f(x)= 1 if x is rational, 0 if x is irrational.pasmith said:
A limit is a mathematical concept that describes the behavior of a function as its input approaches a certain value. It is denoted by the symbol lim and is used to determine the value that a function approaches as its input gets closer and closer to a given value.
A limit exists if the value of the function at the given input approaches a finite number as the input gets closer and closer to the given value. In other words, the function does not have any sudden jumps or holes at the given value.
This notation represents the limit of the quotient of two functions, f(x) and g(x), as x approaches a. It is used to determine if the quotient of the two functions has a finite limit as x approaches a.
Yes, it is possible for lim/x→a/[f(x)g(x)] to exist while lim/x→a/f(x) or lim/x→a/g(x) do not exist. This is because the existence of a limit for a quotient of two functions depends on the behavior of both functions near the given value, rather than just one of the functions.
To show that lim/x→a/[f(x)g(x)] exists, you can use the limit laws and algebraic manipulation to simplify the expression and determine if a finite value can be obtained as x approaches a. Additionally, you can use graphical or numerical methods to visualize and approximate the limit. Finally, you can use the formal definition of a limit to prove its existence.