MHB Can someone please explain why marks were taken off in this limit question

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The discussion centers on the rules of limits in calculus, specifically whether the limit of a product can be separated into the product of the limits. The teacher emphasizes that all limits must be evaluated simultaneously, suggesting that solving them individually may lead to incorrect conclusions. Participants debate the mathematical validity of separating limits, with some arguing that it is only valid under certain conditions. The key question posed is whether the equation lim[f(x)*g(x)] = lim[f(x)]*lim[g(x)] holds true universally. Understanding the conditions under which this equation is valid is crucial for accurate limit evaluation.
FrancisG
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My teacher says I have to solve all the limits at once, but I don't understand why it is mathematically incorrect to solve one limit before the other. The test is here:
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Is lim[f(x)*g(x)] = lim[f(x)]*lim[g(x)] ALWAYS?
 

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