Final answer:
To prove that the function f(x) is not differentiable at the origin, we need to show that the limit of the difference quotient does not exist at x = 0. The difference quotient is defined as f'(0) = lim_(h→0) [(f(h) - f(0))/h]. Evaluating the limit, we find that it does not exist, indicating that the function f(x) is not differentiable at the origin.
Step-by-step explanation:
To prove that the function f(x) is not differentiable at the origin, we need to show that the limit of the difference quotient does not exist at x = 0. The difference quotient is defined as f'(0) = lim_(h→0) [(f(h) - f(0))/h].
Substituting f(x) = x/e^(1/x+1), we get f'(0) = lim_(h→0) [(f(h) - f(0))/h] = lim_(h→0) [(h/e^(1/h+1))/h] = lim_(h→0) [1/e^(1/h+1)].
To evaluate the limit, we need to consider the behavior of e^(1/h+1) as h approaches 0. As h approaches 0 from the right (h > 0), e^(1/h+1) approaches infinity. As h approaches 0 from the left (h < 0), e^(1/h+1) approaches 0. Therefore, the limit of f'(0) does not exist, indicating that the function f(x) is not differentiable at the origin.