Harnessing the Implications of Given Conditions when Evaluating Limits

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Another interesting question that I encountered in my practice is the following. I would like to know how to de-construct the implications of a particular assertion of the question

The following is the question:

The question.

The given solutions are: 17)i) $a^2-1, 0$ ii) $ \pm 1, -1$

I obtained the solution for this question, but I was rather curious about why the question stated $|\text g(x)|\le2$, and then use $\text g({1\over x})$ in the definition of $f(x)$.

I would thus love to hear some thoughts about this.