Trigger said:Would f(x) have to be -x squared if f(1/x) is to be proportional to 1/x
A function f:R-R is a mathematical representation that maps each element in the set of real numbers (R) to another element in the same set. It takes an input value (x) and produces an output value (f(x)) based on a specific rule or equation.
"lim x→0" represents the limit as x approaches 0. In other words, it describes the behavior of a function as the input value (x) gets closer and closer to 0.
If xf(1/x) were to equal 0, it would mean that the output value (f(x)) is equal to 0 for all inputs close to 0. This would indicate that the function has a horizontal asymptote at x=0, which goes against the requirement that the limit at x=0 must not exist for this function.
One example of such a function is f(x) = sin(1/x). As x approaches 0, sin(1/x) oscillates between -1 and 1, so xf(1/x) will also oscillate and never equal 0.
The limit of this function at x=0 can be determined by evaluating the function for different values of x that get closer and closer to 0. If the output values approach a specific number or range of numbers, that would be the limit. However, if the output values do not approach a specific number or range, the limit does not exist.