If 𝑓(𝑥) = { 𝑥sin 1 𝑥 , 𝑥 ≠ 0 0, 𝑥 = 0 , then lim𝑥→0𝑓(𝑥) =

The limit of the function f(x) as x approaches 0 can be found using L'Hopital's Rule, which states that the limit of a quotient of two functions as x approaches a certain value is equal to the limit of the quotients of their derivatives.

First, we need to express f(x) as a quotient of two functions. We can write f(x) = sin(1/x) / (1/x) for x ≠ 0 and 0 for x = 0.

Next, we find the derivatives of the numerator and the denominator. The derivative of sin(1/x) is cos(1/x) * -1/x^2, and the derivative of 1/x is -1/x^2.

So, the limit of f(x) as x approaches 0 is the same as the limit of [cos(1/x) * -1/x^2] / [-1/x^2] as x approaches 0.

The -1/x^2 terms cancel out, leaving us with the limit of cos(1/x) as x approaches 0.

However, as x approaches 0, 1/x approaches infinity, and the cosine function oscillates between -1 and 1 for all real numbers. Therefore, the limit does not exist because it does not approach a specific value.