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Consider the function f(x)=g(x) (e^(1/x)-e^(-1/x))/(e^(1/x)+e^(-1/x)), where g is continuous. The limit lim_(x→ 0) f(x) fails to exist when

1. g(x) is a constant function
2. g(x)=x
3. g(x)=x²
4. g(x)=x h(x), where h(x) is a polynomial

Correct answer: g(x) is a constant function

Solution

The factor (e^(1/x)-e^(-1/x))/(e^(1/x)+e^(-1/x)) = tanh(1/x) tends to +1 as x->0+ and -1 as x->0-. So the limit of f exists only if g(0)=0. If g is a (non-zero) constant, the left and right limits are -g(0) and +g(0), which differ, so the limit fails to exist. For g(x)=x, x^2, or x*h(x) we have g(0)=0 and the limit exists.

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