Question 7.9.6: Let f(x) = 1/x² and g(x) = 1/x^4. As x → 0, so f(x)→∞and g(x......

Essential Mathematics for Economic Analysis [1185377]

Let f (x) = 1/x² and $g(x) = 1/x^{4}.$ As x → 0, so $f (x)\to\infty$ and $g(x)\to\infty.$ Examine the limits as x → 0 of the following expressions:

f(x)-g(x),\ g(x)-f(x),\ {\frac{f(x)}{g(x)}},\ {\mathrm{and}}\ {\frac{g(x)}{f(x)}}

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$f(x)-g(x)=(x^{2}-1)/x^{4}.{\mathrm{As}}\ \ x\to0,$ the numerator tends to −1 and the denominator to 0, so the fraction tends to $-\infty.$ For the other three limits we have:

g(x)-f(x)={\frac{1-x^{2}}{x^{4}}}\to\infty,\,{\frac{f(x)}{g(x)}}=x^{2}\to0,\,{\mathrm{~and~}}{\frac{g(x)}{f(x)}}={\frac{1}{x^{2}}}\to\infty

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