$Q11)\lim_{x\to\infty }(\frac{x+8}{x-2})^{x}$
$Q12)\lim_{x\to\infty }(x+3)(ln(x-2)-ln(x+3))$
$Q13)\lim_{(x,y)\to(2,0)}\frac{sin(xy)}{y}$
$Q14)\lim_{(x,y)\to(0,0)}\frac{xe^{y}-x}{ysinx}$
$Q15)\lim_{x\to 0}\frac{sin(tan(sinx))}{x}$
$Q11)\lim_{x\to\infty }(\frac{x+8}{x-2})^{x}$
$Q12)\lim_{x\to\infty }(x+3)(ln(x-2)-ln(x+3))$
$Q13)\lim_{(x,y)\to(2,0)}\frac{sin(xy)}{y}$
$Q14)\lim_{(x,y)\to(0,0)}\frac{xe^{y}-x}{ysinx}$
$Q15)\lim_{x\to 0}\frac{sin(tan(sinx))}{x}$
; 06-06-2016 03:49 PM
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\[{\color{Red} Q11)}\lim_{x\rightarrow \infty }\left ( \frac{x+8}{x-2} \right )^{x}=\\\lim_{x\rightarrow \infty }\left ( 1+\frac{10}{x-2} \right )^{x}=\\\lim_{x\rightarrow \infty }\left ( 1+\frac{10}{x-2} \right )^{\frac{x-2}{10}\times \frac{10x}{x-2}}=\\\lim_{x\rightarrow \infty }\left ( 1+\frac{10}{x-2} \right )^{\frac{x-2}{10}\times \frac{10}{1-\frac{2}{x}}}={\color{Blue} e^{10}}\]
[URL="http://alnasiry.net/forums"][IMG]http://alnasiry.net/forums/uploaded/2_iraqiflag.gif[/IMG][/URL]
[IMG]$\huge (12){\color{Red} \lim_{x\rightarrow \infty }(x+3)(ln(x-2)-ln(x+3))}=\ln\lim_{x\rightarrow \infty }(\frac{x-2}{x+3})^{x+3}=\ln\lim_{x\rightarrow \infty }(1-\frac{a}{x+3})^{x+3}=ln(e^{-a})=-a$ [/IMG]
15
[IMG]$\huge {\color{Magenta} Q13-\lim_{(x,y)\rightarrow (2,0)}(\frac{sin(xy)}{y})}=\lim_{(x,y)\rightarrow (2,0)}(\frac{\frac{sin(xy)}{xy}}{\frac{y}{xy}})=\f rac{1}{\frac{1}{2}}=2$ [/IMG]
$Q14)\lim_{(x,y)\to(0,0)}\frac{xe^{y}-x}{ysinx}$
$=\lim_{(x,y)\to(0,0)}\frac{x(e^{y}-1)}{(sinx)(y)}$
$=\lim_{(x,y)\to(0,0)}(\frac{e^{y}-1}{y})\frac{x}{sinx}=1.1=1$
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$Q16)\lim_{n\to\infty }(\frac{7}{5}+1+\frac{91}{125}+...+\frac{3^{n}+4^{ n}}{5^{n}})$
$Q17)\lim_{x\to\infty }\frac{3x+\sqrt{3x}}{3x+3sinx}$
$Q18)\lim_{n\to\infty }\frac{(n) \sqrt{1+3+5+...+(2n-1)}}{2n^{2}+n+1}$
$Q19)\lim_{n\to\infty }(\frac{n^{2}+2n+2}{n^{2}+n+1})^{3n-1}$
$Q20)\lim_{n\to\infty }(\frac{2^{n}+3}{2^{n}+1})^{n}$
; 06-09-2016 02:20 PM
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18 n^2 1/2
; 06-08-2016 06:19 PM
$Q21)\lim_{n\to\infty }\arccos (\frac{4n^{2}-4n^{5}}{8n^{5}+18n^{4}})$
$Q22)\lim_{n\to\infty }\arcsin (\frac{\sqrt{n+3}+3}{n+3})$
$Q23)\lim_{x\to0}\frac{\sqrt[6]{4+5cosx}-\sqrt[6]{9}}{cosx-1}$
$Q24)\lim_{x\to0}\frac{(3-2cosx)^{4}-(4cosx-3)^{3}}{cosx-1}$
$Q25)\lim_{x\to0}\frac{xsin5x}{cos^{2}x-cos^{3}x}$
; 06-09-2016 02:21 PM
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; 06-09-2016 09:52 AM
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