 The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x usagirl007A 2021-09-13 Answered

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits. $$\lim_{x \to \infty} \frac{((x^{-1})+(x^{-4}))}{((x^{-2})-(x^{-3}))}$$

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$$\displaystyle{x}^{{-{{2}}}}$$ is highest power of x in denominator. Dividing with $$x^{-2}$$ is same as multiplying numerator and denominator with $$\displaystyle{x}^{{2}}$$. ​

$$\lim_{x \to \infty} \frac{(x^{-1})+(x^{-4}))}{((x^{-2})−(x^{-3}))} \cdot \frac{(x^2)}{(x^2)}=\lim_{x \to \infty} \frac{(x+x^{-2})}{(1-x^{-1})}=$$

Use that $$\lim_{x \to \infty} x^{-n}=0 = \frac{(\infty+0)}{(1-0)}=\infty$$