Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/197

Illustrative Example 1. Evaluate ${\tfrac {f(x)}{F(x)}}={\tfrac {x^{3}-3x+2}{x^{3}-x^{2}-x-1}}$ when x = 1.

Solution.	${\frac {f(1)}{F(1)}}=\left.{\frac {x^{3}-3x+2}{x^{3}-x^{2}+1}}\right]_{x=1}={\frac {1-3+2}{1-1-1+1}}={\frac {0}{0}}.$ ∴ indeterminate.
	${\frac {f'(1)}{F'(1)}}=\left.{\frac {3x^{2}-3}{3x^{2}-2x-1}}\right]_{x=1}={\frac {3-3}{3-2-1}}={\frac {0}{0}}.$ ∴ indeterminate.
	${\frac {f''(1)}{F''(1)}}=\left.{\frac {6x}{6x-2}}\right]_{x=1}={\frac {6}{6-2}}={\frac {3}{2}}.$ Ans.

Illustrative Example 2. Evaluate $\lim _{x\to 0}{\frac {e^{x}-e^{-x}-2x}{x-\sin x}}.$

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

EXAMPLES

Evaluate the following by differentiation:^[1]

1. $\lim _{x\to 4}{\frac {x^{2}-16}{x^{2}+x-20}}.$	Ans.	${\frac {8}{9}}$.	9. $\lim _{\theta \to 0}{\frac {\theta -\arcsin \theta }{\sin ^{3}\theta }}.$	Ans.	$-{\frac {1}{6}}.$
2. $\lim _{x\to 1}{\frac {x-1}{x^{n}-1}}.$		${\frac {1}{n}}.$	10. $\lim _{x\to \phi }{\frac {\sin x-\sin \phi }{x-\phi }}.$		$\cos \phi$
3. $\lim _{x\to 1}{\frac {\log x}{x-1}}$		1.	11. $\lim _{y\to 0}{\frac {e^{y}+\sin y-1}{\log(1+y)}}.$		2.
4. $\lim _{x\to 0}{\frac {e^{x}-e^{-x}}{\sin x}}.$		2.	12. $\lim _{\theta \to 0}{\frac {\tan \theta +\sec \theta -1}{\tan \theta -\sec \theta +1}}.$		1.
5. $\lim _{\frac {\tan x-x}{x-\sin x}}.$		2.	13. $\lim _{\phi \to {\frac {\pi }{4}}}{\frac {\sec ^{2}\phi -2\tan \phi }{1+\cos 4\phi }}.$		${\frac {1}{2}}.$
6. $\lim _{x\to {\frac {\pi }{2}}}{\frac {\log \sin x}{(\pi -2x)^{2}}}.$		$-{\frac {1}{8}}.$	14. $\lim _{z\to a}{\frac {az-z^{2}}{a^{4}-2a^{3}z+2az^{3}-z^{4}}}.$		$+\infty .$
7. $\lim _{x\to 0}{\frac {a^{x}-b^{x}}{x}}.$		$\log {\frac {a}{b}}.$	15. $\lim _{x\to 2}{\frac {(e^{x}-e^{2})^{2}}{(x-4)e^{x}+e^{2}x}}.$		$6e^{4}.$
8. $\lim _{r\to a}{\frac {r^{3}-ar^{2}-a^{2}r+a^{3}}{r^{2}-a^{2}}}.$		0.

16. $\lim _{x\to 1}{\frac {x^{2}+x-2}{x^{2}-1}}.$	18. $\lim _{x\to 0}{\frac {\sin 2x}{x}}.$	20. $\lim _{x\to 1}{\frac {\log \cos(x-1)}{1-\sin {\frac {\pi x}{2}}}}.$
17. $\lim _{x\to -2}{\frac {x^{3}+8}{x^{5}+32}}.$	19. $\lim _{x\to 0}{\frac {x-\sin x}{x^{3}}}.$	21. $\lim _{x\to 0}{\frac {\tan x-\sin x}{\sin ^{3}x}}.$

1. After differentiating, the student should in every case reduce the resulting expression to its simplest possible form before substituting the value of the variable.