Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/200



Given a function of the form

Evaluating this by the process illustrated in §113 gives the limit of the logarithm of the function. This being equal to the logarithm of the limit of the function, the limit of the function is known.

Evaluate $$x^x$$ when $$x = 0$$.


 * Solution. This function assumes the indeterminate form $$0^0$$ for $$x = 0$$.


 * {| style="width: 100%;"


 * Let
 * style="text-align: right;"|$$\ y$$
 * colspan="2"|$$=\ x^x;$$
 * then
 * style="text-align: right;"|$$\ \log y$$
 * $$= x \log x = 0 \cdot -\infty,$$
 * when $$x = 0$$.
 * By § 113,
 * style="text-align: right;"|$$\ \log y$$
 * $$\frac{\log x}{\frac{1}{x}} = \frac{-\infty}{\infty},$$
 * when $$x = 0$$.
 * }
 * when $$x = 0$$.
 * }