Find functions f and g such that lim x→1 f(x)=0 and lim x→1 (f(x)g(x))=5.

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Consider the limit condition: \( \lim_{x \to 1} f(x) = 0 \). This implies that as \( x \) approaches 1, \( f(x) \) approaches 0.

To satisfy \( \lim_{x \to 1} (f(x)g(x)) = 5 \), we need \( g(x) \) to behave in such a way that the product \( f(x)g(x) \) approaches 5 as \( x \) approaches 1.

One possible approach is to let \( f(x) = (x-1) \), which clearly satisfies \( \lim_{x \to 1} f(x) = 0 \).

Now, choose \( g(x) = \frac{5}{x-1} \). This choice ensures that the product \( f(x)g(x) = (x-1) \cdot \frac{5}{x-1} = 5 \) for all \( x \neq 1 \).

Verify that \( \lim_{x \to 1} (f(x)g(x)) = \lim_{x \to 1} 5 = 5 \), which satisfies the given condition.

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