what is
 
 
1.lim (x/x+3)power(3x)
(x-->infinte)
 
2.lim {(x+6)/(x+4)}power(x+4)
  (x->infinite)
3. lim (e)power(-1/x)
 (x->0)
 
equal to????
and is there any trick to solve this type of que in competition exam.

Asked by Vishwajeet Rai | 22nd Jul, 2014, 02:05: PM

Expert Answer:

limit as x rightwards arrow infinity of open parentheses fraction numerator x over denominator x plus 3 end fraction close parentheses to the power of 3 x end exponent equals e to the power of limit as x rightwards arrow infinity of 3 x. log open parentheses fraction numerator x over denominator x plus 3 end fraction close parentheses end exponent equals e to the power of limit as x rightwards arrow infinity of 3 x. log open parentheses 1 plus fraction numerator x over denominator x plus 3 end fraction minus 1 close parentheses end exponent equals e to the power of limit as x rightwards arrow infinity of 3 x. log open parentheses 1 plus fraction numerator x minus x minus 3 over denominator x plus 3 end fraction close parentheses end exponent equals e to the power of limit as x rightwards arrow infinity of 3 x. log open parentheses 1 minus fraction numerator 3 over denominator x plus 3 end fraction close parentheses end exponent equals e to the power of limit as x rightwards arrow infinity of 3 x. fraction numerator log open parentheses 1 minus fraction numerator 3 over denominator x plus 3 end fraction close parentheses over denominator begin display style fraction numerator minus 3 over denominator left parenthesis x plus 3 right parenthesis end fraction end style end fraction. fraction numerator minus 3 over denominator left parenthesis x plus 3 right parenthesis end fraction end exponent equals e to the power of limit as x rightwards arrow infinity of fraction numerator log open parentheses 1 minus fraction numerator 3 over denominator x plus 3 end fraction close parentheses over denominator begin display style fraction numerator minus 3 over denominator left parenthesis x plus 3 right parenthesis end fraction end style end fraction. limit as x rightwards arrow infinity of fraction numerator minus 9 x over denominator left parenthesis x plus 3 right parenthesis end fraction end exponent equals e to the power of limit as fraction numerator minus 3 over denominator x plus 3 end fraction rightwards arrow 0 of fraction numerator log open parentheses 1 minus fraction numerator 3 over denominator x plus 3 end fraction close parentheses over denominator begin display style fraction numerator minus 3 over denominator left parenthesis x plus 3 right parenthesis end fraction end style end fraction. limit as x rightwards arrow infinity of fraction numerator minus 9 x over denominator left parenthesis x plus 3 right parenthesis end fraction end exponent open square brackets I f space x rightwards arrow infinity comma space t h e n space fraction numerator minus 1 over denominator x plus 3 end fraction rightwards arrow 0 space & fraction numerator minus 3 over denominator x plus 3 end fraction rightwards arrow 0 space close square brackets  equals e to the power of open square brackets limit as fraction numerator minus 3 over denominator x plus 3 end fraction rightwards arrow 0 of fraction numerator log open parentheses 1 minus fraction numerator 3 over denominator x plus 3 end fraction close parentheses over denominator begin display style fraction numerator minus 3 over denominator left parenthesis x plus 3 right parenthesis end fraction end style end fraction close square brackets. limit as x rightwards arrow infinity of fraction numerator minus 9 x over denominator left parenthesis x plus 3 right parenthesis end fraction end exponent equals e to the power of limit as fraction numerator minus 3 over denominator x plus 3 end fraction rightwards arrow 0 of fraction numerator log open parentheses 1 minus fraction numerator 3 over denominator x plus 3 end fraction close parentheses over denominator begin display style fraction numerator minus 3 over denominator left parenthesis x plus 3 right parenthesis end fraction end style end fraction. limit as x rightwards arrow infinity of fraction numerator minus 9 x over denominator left parenthesis x plus 3 right parenthesis end fraction open square brackets limit as x rightwards arrow 0 of fraction numerator log left parenthesis 1 plus x right parenthesis over denominator x end fraction equals 1 close square brackets end exponent equals e to the power of 1 cross times limit as x rightwards arrow infinity of fraction numerator minus 9 x over denominator left parenthesis x plus 3 right parenthesis end fraction end exponent equals e to the power of limit as x rightwards arrow infinity of fraction numerator minus 9 over denominator left parenthesis 1 plus begin display style 3 over x end style right parenthesis end fraction end exponent equals e to the power of fraction numerator minus 9 over denominator left parenthesis 1 plus begin display style 3 over infinity end style right parenthesis end fraction end exponent equals e to the power of fraction numerator minus 9 over denominator left parenthesis 1 plus begin display style 0 end style right parenthesis end fraction end exponent equals e to the power of minus 9 end exponent  T h e space b e s t space w a y space i s space t o space t a k e space log space o n space b o t h space s i d e s space f o r space t h e s e space s u m s.

Answered by Prasenjit Paul | 22nd Jul, 2014, 05:26: PM