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1) Умножим числитель и знаменатель на ![sqrt{x+3}+1 sqrt{x+3}+1](https://tex.z-dn.net/?f=+sqrt%7Bx%2B3%7D%2B1+)
![lim_{x to inft-2} frac{(sqrt{x+3}-1)*(sqrt{x+3}+1)}{(x^{2} -4)*(sqrt{x+3}+1)} = lim_{x to inft-2} frac{x+2}{(x+2)*(x-2)*(sqrt{x+3}+1)} =
=lim_{x to inft-2}frac{1}{(x-2)*(sqrt{x+3}+1)}=frac{1}{(-2-2)*( sqrt{-2+3}+1 )} =- frac{1}{8} lim_{x to inft-2} frac{(sqrt{x+3}-1)*(sqrt{x+3}+1)}{(x^{2} -4)*(sqrt{x+3}+1)} = lim_{x to inft-2} frac{x+2}{(x+2)*(x-2)*(sqrt{x+3}+1)} =
=lim_{x to inft-2}frac{1}{(x-2)*(sqrt{x+3}+1)}=frac{1}{(-2-2)*( sqrt{-2+3}+1 )} =- frac{1}{8}](https://tex.z-dn.net/?f=lim_%7Bx+to+inft-2%7D+frac%7B%28sqrt%7Bx%2B3%7D-1%29%2A%28sqrt%7Bx%2B3%7D%2B1%29%7D%7B%28x%5E%7B2%7D+-4%29%2A%28sqrt%7Bx%2B3%7D%2B1%29%7D+%3D+lim_%7Bx+to+inft-2%7D+frac%7Bx%2B2%7D%7B%28x%2B2%29%2A%28x-2%29%2A%28sqrt%7Bx%2B3%7D%2B1%29%7D+%3D%0A%0A%3Dlim_%7Bx+to+inft-2%7Dfrac%7B1%7D%7B%28x-2%29%2A%28sqrt%7Bx%2B3%7D%2B1%29%7D%3Dfrac%7B1%7D%7B%28-2-2%29%2A%28+sqrt%7B-2%2B3%7D%2B1+%29%7D+%3D-+frac%7B1%7D%7B8%7D)
2) Числитель и знаменатель первой дроби разделим на n²
![lim_{n to infty} frac{ n^{2}+2n+1 }{n^{2}-n+2}- frac{n}{3} = lim_{n to infty} frac{1+ frac{2}{n}+ frac{1}{n^{2}}}{1- frac{1}{n}+ frac{2}{n}} - frac{n}{3} = lim_{n to infty} frac{ n^{2}+2n+1 }{n^{2}-n+2}- frac{n}{3} = lim_{n to infty} frac{1+ frac{2}{n}+ frac{1}{n^{2}}}{1- frac{1}{n}+ frac{2}{n}} - frac{n}{3} =](https://tex.z-dn.net/?f=+lim_%7Bn+to+infty%7D++frac%7B+n%5E%7B2%7D%2B2n%2B1+%7D%7Bn%5E%7B2%7D-n%2B2%7D-+frac%7Bn%7D%7B3%7D+%3D+lim_%7Bn+to+infty%7D++frac%7B1%2B+frac%7B2%7D%7Bn%7D%2B+frac%7B1%7D%7Bn%5E%7B2%7D%7D%7D%7B1-+frac%7B1%7D%7Bn%7D%2B+frac%7B2%7D%7Bn%7D%7D+-+frac%7Bn%7D%7B3%7D+%3D)
=![frac{1+ frac{2}{oo}+ frac{1}{oo^{2}}}{1- frac{1}{oo}+ frac{2}{oo}} - frac{oo}{3}=1-oo=-oo frac{1+ frac{2}{oo}+ frac{1}{oo^{2}}}{1- frac{1}{oo}+ frac{2}{oo}} - frac{oo}{3}=1-oo=-oo](https://tex.z-dn.net/?f=frac%7B1%2B+frac%7B2%7D%7Boo%7D%2B+frac%7B1%7D%7Boo%5E%7B2%7D%7D%7D%7B1-+frac%7B1%7D%7Boo%7D%2B+frac%7B2%7D%7Boo%7D%7D+-+frac%7Boo%7D%7B3%7D%3D1-oo%3D-oo)
3) Числитель и знаменатель умножим на![e^{ x^{2} } e^{ x^{2} }](https://tex.z-dn.net/?f=+e%5E%7B+x%5E%7B2%7D+%7D+)
![lim_{x to inft0} frac{ e^{ x^{2}}*ln(1+ x^{2} )}{ e^{ x^{2}}*( e^{ -x^{2}}-1 )} = lim_{x to inft0} e^{ x^{2}}*lim_{x to inft0} frac{ln(1+x^{2})}{(1- e^{ x^{2}})} =
=1*lim_{x to inft0} - frac{ln(1+ x^{2} )}{ e^{ x^{2} -1}} = -lim_{x to inft0} frac{ frac{ln(1+ x^{2})}{x^{2}}} { frac{( e^{ x^{2}}-1 )}{x^{2}}} = lim_{x to inft0} frac{ e^{ x^{2}}*ln(1+ x^{2} )}{ e^{ x^{2}}*( e^{ -x^{2}}-1 )} = lim_{x to inft0} e^{ x^{2}}*lim_{x to inft0} frac{ln(1+x^{2})}{(1- e^{ x^{2}})} =
=1*lim_{x to inft0} - frac{ln(1+ x^{2} )}{ e^{ x^{2} -1}} = -lim_{x to inft0} frac{ frac{ln(1+ x^{2})}{x^{2}}} { frac{( e^{ x^{2}}-1 )}{x^{2}}} =](https://tex.z-dn.net/?f=+lim_%7Bx+to+inft0%7D++frac%7B+e%5E%7B+x%5E%7B2%7D%7D%2Aln%281%2B+x%5E%7B2%7D+%29%7D%7B+e%5E%7B+x%5E%7B2%7D%7D%2A%28+e%5E%7B+-x%5E%7B2%7D%7D-1+%29%7D+%3D+lim_%7Bx+to+inft0%7D+e%5E%7B+x%5E%7B2%7D%7D%2Alim_%7Bx+to+inft0%7D+frac%7Bln%281%2Bx%5E%7B2%7D%29%7D%7B%281-+e%5E%7B+x%5E%7B2%7D%7D%29%7D+%3D%0A%0A%3D1%2Alim_%7Bx+to+inft0%7D+-+frac%7Bln%281%2B+x%5E%7B2%7D+%29%7D%7B+e%5E%7B+x%5E%7B2%7D+-1%7D%7D+%3D+-lim_%7Bx+to+inft0%7D+frac%7B+frac%7Bln%281%2B+x%5E%7B2%7D%29%7D%7Bx%5E%7B2%7D%7D%7D+%7B+frac%7B%28+e%5E%7B+x%5E%7B2%7D%7D-1+%29%7D%7Bx%5E%7B2%7D%7D%7D+%3D)
=![- frac{lim_{x to inft0} frac{ln(1+ x^{2})}{ x^{2} } } {lim_{x to inft0} frac{( e^{ x^{2} }-1 )}{ x^{2} } } =- frac{1}{1} =-1 - frac{lim_{x to inft0} frac{ln(1+ x^{2})}{ x^{2} } } {lim_{x to inft0} frac{( e^{ x^{2} }-1 )}{ x^{2} } } =- frac{1}{1} =-1](https://tex.z-dn.net/?f=-+frac%7Blim_%7Bx+to+inft0%7D+frac%7Bln%281%2B+x%5E%7B2%7D%29%7D%7B+x%5E%7B2%7D+%7D++%7D+%7Blim_%7Bx+to+inft0%7D++frac%7B%28+e%5E%7B+x%5E%7B2%7D+%7D-1+%29%7D%7B+x%5E%7B2%7D+%7D+%7D+%3D-+frac%7B1%7D%7B1%7D+%3D-1)
Разделив числитель и знаменатель ещё на x², привели к следствиям второго замечательного предела.
4) Числитель и знаменатель делим на x²
2) Числитель и знаменатель первой дроби разделим на n²
=
3) Числитель и знаменатель умножим на
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Разделив числитель и знаменатель ещё на x², привели к следствиям второго замечательного предела.
4) Числитель и знаменатель делим на x²
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