Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/48

 2. $\scriptscriptstyle {{\underset {x=\infty }{\operatorname {limit} }}\left({\frac {x^{2}+2x}{5-3x^{2}}}\right)=-{\frac {1}{3}}}$.

Proof.	$\scriptscriptstyle {{\underset {x=\infty }{\operatorname {limit} }}\left({\frac {x^{2}+2x}{5-3x^{2}}}\right)={\underset {x=\infty }{\operatorname {limit} }}\left({\frac {1+{\frac {2}{x}}}{{\frac {5}{x^{2}}}-3}}\right)}$
	[Dividing both numerator and denominator by $\scriptscriptstyle {x^{2}}$.]
	$\scriptscriptstyle {={\frac {{\underset {x=\infty }{\operatorname {limit} }}\left(1+{\frac {2}{x}}\right)}{{\underset {x=\infty }{\operatorname {limit} }}\left({\frac {5}{x^{2}}}-3\right)}}}$	Th. III, p. 18
	$\scriptscriptstyle {={\frac {{\underset {x=\infty }{\operatorname {limit} }}(1)+{\underset {x=\infty }{\operatorname {limit} }}\left({\frac {2}{x}}\right)}{{\underset {x=\infty }{\operatorname {limit} }}\left({\frac {5}{x^{2}}}\right)-{\underset {x=\infty }{\operatorname {limit} }}(3)}}}$	Th. I, p. 18
	$\scriptscriptstyle {={\frac {1+0}{0-3}}=-{\frac {1}{3}}.}$

 3. $\scriptscriptstyle {{\underset {x=1}{\operatorname {limit} }}{\frac {x^{2}-2x+5}{x^{2}+7}}={\frac {1}{2}}}$.

 4. $\scriptscriptstyle {{\underset {x=0}{\operatorname {limit} }}{\frac {3x^{3}+6x^{2}}{2x^{4}-15x^{2}}}=-{\frac {2}{5}}}$.

 5. $\scriptscriptstyle {{\underset {x=-2}{\operatorname {limit} }}{\frac {x^{2}+1}{x+3}}=5}$.

 6. $\scriptscriptstyle {{\underset {h=0}{\operatorname {limit} }}(3ax^{2}-2hx+5h^{2})=3ax^{2}}$.

 7. $\scriptscriptstyle {{\underset {x=\infty }{\operatorname {limit} }}(ax^{2}+bx+c)=\infty }$.

 8. $\scriptscriptstyle {{\underset {k=0}{\operatorname {limit} }}{\frac {(x-k)^{2}-2kx^{3}}{x(x+k)}}=1}$.

 9. $\scriptscriptstyle {{\underset {x=\infty }{\operatorname {limit} }}{\frac {x^{2}+1}{3x^{2}+2x-1}}={\frac {1}{3}}}$.

10. $\scriptscriptstyle {{\underset {x=\infty }{\operatorname {limit} }}{\frac {3+2x}{x^{2}-5x}}=0}$.

11. $\scriptscriptstyle {{\underset {\alpha ={\frac {\pi }{2}}}{\operatorname {limit} }}{\frac {\cos(\alpha -a)}{\cos(2\alpha -a)}}=-\tan a}$.

12. $\scriptscriptstyle {{\underset {x=\infty }{\operatorname {limit} }}{\frac {ax^{2}+bx+c}{dx^{2}+ex+f}}={\frac {a}{d}}}$.

13. $\scriptscriptstyle {{\underset {z=0}{\operatorname {limit} }}{\frac {a}{2}}(e^{\frac {z}{a}}+e^{-{\frac {z}{a}}})=a}$.

14. $\scriptscriptstyle {{\underset {x=0}{\operatorname {limit} }}{\frac {2x^{3}+3x^{2}}{x^{3}}}=\infty }$.

15. $\scriptscriptstyle {{\underset {x=\infty }{\operatorname {limit} }}{\frac {5x^{2}-2x}{x}}=\infty }$.

16. $\scriptscriptstyle {{\underset {y=\infty }{\operatorname {limit} }}{\frac {y}{y+1}}=1}$.

17. $\scriptscriptstyle {{\underset {n=\infty }{\operatorname {limit} }}{\frac {n(n+1)}{(n+2)(n+3)}}=1}$.

18. $\scriptscriptstyle {{\underset {s=1}{\operatorname {limit} }}{\frac {s^{3}-1}{s-1}}=3}$.

19. $\scriptscriptstyle {{\underset {h=0}{\operatorname {limit} }}{\frac {(x+h)^{n}-x^{n}}{h}}=nx^{n-1}}$.

20. $\scriptscriptstyle {{\underset {h=0}{\operatorname {limit} }}\left[\cos(\theta +h){\frac {\sin h}{h}}\right]=\cos \theta }$.

21. $\scriptscriptstyle {{\underset {x=\infty }{\operatorname {limit} }}{\frac {4x^{2}-x}{4-3x^{2}}}=-{\frac {4}{3}}}$.

22. $\scriptscriptstyle {{\underset {\theta =0}{\operatorname {limit} }}{\frac {1-\cos \theta }{\theta ^{2}}}={\frac {1}{2}}}$.

23. $\scriptscriptstyle {{\underset {x=a}{\operatorname {limit} }}{\frac {1}{x-a}}=-\infty }$, if $\scriptscriptstyle {x}$ is increasing as it approaches the value $\scriptscriptstyle {a}$.

24. $\scriptscriptstyle {{\underset {x=a}{\operatorname {limit} }}{\frac {1}{x-a}}=+\infty }$, if $\scriptscriptstyle {x}$ is decreasing as it approaches the value $\scriptscriptstyle {a}$.