If f(x) and g(x) are such that lim f(x) as x --> a = + infinity and lim g(x) as x --> a = 0 Then
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lim [ f(x) . g(x) ] as x --> a is always equal to 0
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lim [ f(x) . g(x) ] as x --> a is never equal to 0
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lim [ f(x) . g(x) ] as x --> a may be +infinity or -infinity
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lim [ f(x) . g(x) ] as x --> a may be equal to a finite value
When one function approaches infinity and another approaches zero, their product is an indeterminate form of type ∞·0. The limit could converge to any finite value, diverge to +∞ or -∞, or equal zero depending on how rapidly each function approaches its limit. Option C is correct because the product may indeed be infinite (with sign depending on the sign of g(x)). Option D is correct because the product may converge to a finite non-zero value (e.g., f(x)=1/x², g(x)=x² as x→0 gives limit 1). Option A is incorrect because it's not always 0. Option B is incorrect because it can be 0 (e.g., f(x)=1/x, g(x)=x as x→0).
When one factor blows up to infinity and the other shrinks to zero, the product is an indeterminate form (infinity * 0), so its actual limiting behavior depends entirely on how fast each piece approaches its limit. Depending on the specific functions, the product can diverge to +infinity or -infinity (e.g., if g shrinks too slowly relative to f), or it can settle on a finite nonzero value (e.g., if f(x)=1/x^2 and g(x)=3x as x approaches a suitable point) — so both of these outcomes are genuinely possible, and no single fixed value is guaranteed.