Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/201

By § 112,

$\ \log y$

${\frac {\frac {1}{x}}{-{\frac {1}{x^{2}}}}}=-x=0,$

when $x=0$.

Since $y=x^{x}$, this gives $log_{e}x^{x}=0$; i.e., $x^{x}=1$. Ans.

Illustrative Example 2. Evaluate $(1+x)^{\tfrac {1}{x}}$; when $x=0$.

Solution. This function assumes the indeterminate form $1^{\infty }$ for $x=0$.

Let	$\ y$	$=(1+x)^{\frac {1}{x}};$
then	$\ \log y$	$={\frac {1}{x}}\log(1+x)=\infty \cdot .0,$	when $x=0$.
By § 113,	$\ y$	$={\frac {\log(1+x)}{x}}={\frac {0}{0}},$	when x = 0.
By § 111,	$\ y$	$={\frac {\frac {1}{1+x}}{1}}={\frac {1}{1+x}}=1,$	when $x=0$.

since $y=(1+x)^{\frac {1}{1+x}}$, this gives $log_{e}(1+x)^{\frac {1}{x}}=1$; i.e. $(1+x)^{\frac {1}{x}}=e$. Ans.

Illustrative Example 3. Evaluate $(\cot x)^{\sin x}$ for $x=0$.

Solution. This function assumes the indeterminate form $\infty ^{0}$ for $x=0$.

Let	$\ y$	$=\ (\cot x)^{\sin x};$
then	$\ \log y$	$=\ \sin x\log \cot x=0\cdot \infty ,$	when $x=0$.
By § 113,	$\ \log y$	$={\frac {\log \cot x}{\csc x}}={\frac {\infty }{\infty }},$	when $x=0$.
By § 112,	$\ \log y$	$={\frac {\frac {-\csc ^{2}x}{\cot x}}{-\csc x\cot x}}={\frac {\sin x}{\cos ^{2}x}}=0,$	when x = 0.

EXAMPLES

Evaluate the following expressions by differentiation:

1. $\lim _{x\to 1}x^{\frac {1}{1-x}}.$	Ans.	${\frac {1}{e}}.$	7. $\lim _{x\to 0}(e^{x}+x)^{\frac {1}{x}}.$	Ans.	$e^{2}.$
2. $\lim _{x\to 0}\left({\frac {1}{x}}\right)^{\tan x}.$		1.	8. $\lim _{x\to 0}(\cot x)^{\frac {1}{\log x}}.$		${\frac {1}{e}}.$
3. $\lim _{\theta \to {\frac {\pi }{2}}}(\sin \theta )^{\tan \theta }.$		1.	9. $\lim _{z\to 0}(1+nz)^{\frac {1}{z}}.$		$e^{n}.$
4. $\lim _{y\to \infty }\left(1+{\frac {a}{y}}\right)^{y}.$		$e^{a}.$	10. $\lim _{\phi \to 1}\left(\tan {\frac {\pi \phi }{4}}\right)^{\tan {\frac {\pi \phi }{2}}}.$		${\frac {1}{e}}.$
5. $\lim _{x\to 0}(1+\sin x)^{\cot x}.$		e.	11. $\lim _{\theta \to 0}(\cos m\theta )^{\frac {n}{\theta ^{2}}}.$		$e^{-{\frac {1}{2}}nm^{2}}.$
6. $\lim _{x\to \infty }\left({\frac {2}{x}}+1\right)^{x}.$		$e^{2}.$	12. $\lim _{x\to 0}(\cot x)^{x}.$		1.
			13. $\lim _{x\to a}\left(2-{\frac {x}{a}}\right)^{\tan {\frac {\pi x}{2a}}}.$		$e^{\frac {2}{\pi }}.$

---