Expressions assuming the form $\frac {\infty }{\infty }$

As $\infty$ is not a number, the expression $\infty\ \div\ \infty$ is indeterminate. To evaluate a fraction assuming this form, the numerator and denominator being algebraic functions, we shall find useful the following

RULE. Divide both numerator and denominator by the highest power of the variable occurring in either. Then substitute the value of the variable.

Example 3.12.1   Evaluate

Solution. Substituting directly, we get

$$\lim_{x \to \infty} \frac{2x^3 - 3x^2 + 4}{5x - x^2 - 7x^3} = \frac{\infty}{\infty}$$

which is indeterminate. Hence, following the above rule, we divide both numerator and denominator by $x^3$, Then

$$\lim_{x \to \infty} \frac{2x^3 - 3x^2 + 4}{5x - x^2 - 7x^3} = \l... ...\frac{3}{x} + \frac{4}{x^3}}{\frac{5}{x^2} - \frac{1}{x} - 7} = -\frac{2}{7}.$$