Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/199

Illustrative Example 1. Evaluate $\sec 3x\cos 5x$ for $x={\tfrac {\pi }{2}}$.

Solution. $\left.\sec 3x\cos 5x\right]_{x={\frac {\pi }{2}}}=\infty \cdot 0.$ ∴ indeterminate.

Substituting ${\tfrac {1}{\cos 3x}}$ for $\sec 3x$, the function becomes ${\tfrac {\cos 5x}{\cos 3x}}={\tfrac {f(x)}{F(x)}}$.

${\frac {f({\frac {\pi }{2}})}{F({\frac {\pi }{2}})}}$	$=\left.{\frac {\cos 5x}{\cos 3x}}\right]_{x={\frac {\pi }{s}}}={\frac {0}{0}}.$ indeterminate.
${\frac {f'({\frac {\pi }{2}})}{F'({\frac {\pi }{2}})}}$	$=\left.{\frac {-\cos x}{-\sin x}}\right]_{x={\frac {\pi }{s}}}={\frac {0}{-1}}=0.$ Ans.

114. Evaluation of the indeterminate form $\infty -\infty$. It is possible in general to transform the expression into a fraction which will assume either the form ${\tfrac {0}{0}}$ or ${\tfrac {\infty }{\infty }}$.

Illustrative Example 1. Evaluate $\sec x-\tan x$ for $x={\tfrac {\pi }{2}}$.

Solution. $\left.\sec x-\tan x\right]_{x={\tfrac {\pi }{2}}}=\infty -\infty .$ ∴ indeterminate.

By Trigonometry, $\sec x-\tan x={\tfrac {1}{\cos x}}-{\tfrac {\sin x}{\cos x}}={\tfrac {1-\sin x}{\cos x}}={\tfrac {f(x)}{F(x)}}$.

${\frac {f({\frac {\pi }{2}})}{F({\frac {\pi }{2}})}}$	$=\left.{\frac {1-\sin x}{\cos x}}\right]_{x={\frac {\pi }{2}}}={\frac {1-1}{0}}={\frac {0}{0}}.$ ∴ indeterminate.
${\frac {f'({\frac {\pi }{2}})}{F'({\frac {\pi }{2}})}}$	$=\left.{\frac {-\cos x}{-\sin x}}\right]_{x={\frac {\pi }{2}}}={\frac {0}{-1}}=0.$ Ans.

EXAMPLES

Evaluate the following expressions by differentiation:^[1]

1. $\lim _{x\to \infty }{\frac {ax^{2}+b}{cx^{2}+d}}.$	Ans.	${\frac {a}{c}}.$	6. $\lim _{x\to 0}{\frac {\log \sin 2x}{\log \sin x}}.$	Ans.	1.
2. $\lim _{x\to 0}{\frac {\cot x}{\log x}}.$		$-\infty .$	7. $\lim _{\theta \to {\frac {\pi }{2}}}{\frac {\tan \theta }{\tan 3\theta }}.$		3.
3. $\lim _{x=\infty }{\frac {\log x}{x^{n}}}.$		0.	8. $\lim _{\phi \to {\frac {\pi }{2}}}{\frac {\log \left(\phi -{\frac {\pi }{2}}\right)}{\tan \phi }}.$		0.
4. $\lim _{x\to \infty }{\frac {x^{2}}{e^{x}}}.$		0.	9. $\lim _{x\to 0}{\frac {\log x}{\cot x}}.$		0.
5. $\lim _{x\to \infty }{\frac {e^{x}}{\log x}}.$		$\infty .$	10. $\lim _{x\to 0}x\log \sin x.$		0.

1. In solving the remaining examples in this chapter it may be of assistance to the student to refer to §24, where many special forms not indeterminate are evaluated.