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If f(x)=sgn(x)" and "g(x)=x^(3),then pro...

If `f(x)=sgn(x)" and "g(x)=x^(3)`,then prove that `lim_(xto0) f(x).g(x)` exists though `lim_(xto0) f(x)` does not exist.

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To prove that \( \lim_{x \to 0} f(x) \cdot g(x) \) exists while \( \lim_{x \to 0} f(x) \) does not exist, we will analyze the functions \( f(x) = \text{sgn}(x) \) and \( g(x) = x^3 \). ### Step-by-Step Solution: 1. **Understanding the Functions**: - The signum function \( f(x) = \text{sgn}(x) \) is defined as: \[ f(x) = ...
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