Q. 21 - Calculus

Question

${\infty }^0$ that approaches (a) $1,$ (b) $2,$ (c) $\infty .$

Step-by-Step Solution

Answer

${\infty }^0$ approaches to (a) $1.$

Step 1. Given information

The given information is ${\infty }^0$, we need to write ${\infty }^0$ approaches to (a) $1,$ (b) $2,$ (c) $\infty .$

Step 2. The explanation for the correct option

For example, $\underset{x\to \infty }{\lim }\left(x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$x$}\right.}\right).$

                     $=\underset{x\to \infty }{\lim }{\left(e^{\ln x}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$x$}\right.}.$

                     $=\underset{x\to \infty }{\lim }\left(e^{\raisebox{1ex}{$\ln x$}\!\left/ \!\raisebox{-1ex}{$x$}\right.}\right).$

                     $=e\left(\underset{x\to \infty }{\lim }{e}^{\frac{\ln x}{x}}\right).$

                      $=e^{\underset{x\to \infty }{\lim }\frac{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$x$}\right.}}{1}.}$ (Applying the L-Hospital rule).

                      $=e^0.$

                       $=1.$