limits

Asked by pritamsadhu99 | 1st Jun, 2020, 01:01: AM

Expert Answer:

To find: lim x → 0 (ex+e-x-2/x2)^{1/x2}
L e t space
L equals limit as x rightwards arrow 0 of open parentheses fraction numerator e to the power of x plus e to the power of negative x end exponent minus 2 over denominator x squared end fraction close parentheses to the power of 1 over x squared end exponent
equals limit as x rightwards arrow 0 of open parentheses fraction numerator e to the power of 2 x end exponent minus 2 e to the power of x plus 1 over denominator e to the power of x x squared end fraction close parentheses to the power of 1 over x squared end exponent
equals limit as x rightwards arrow 0 of open square brackets open parentheses fraction numerator e to the power of x minus 1 over denominator x end fraction close parentheses squared cross times 1 over e to the power of x close square brackets to the power of 1 over x squared end exponent space space... space T h i s space i s space o f space t h e space f o r m space 1 to the power of infinity
equals limit as x rightwards arrow 0 of space 1 over x squared open square brackets open parentheses fraction numerator e to the power of x minus 1 over denominator x end fraction close parentheses squared cross times 1 over e to the power of x minus 1 close square brackets space... space S i n c e space limit as x rightwards arrow 0 of f left parenthesis x right parenthesis to the power of g left parenthesis x right parenthesis end exponent equals stack lim with x rightwards arrow 0 below g left parenthesis x right parenthesis open square brackets f left parenthesis x right parenthesis minus 1 close square brackets
equals limit as x rightwards arrow 0 of space e to the power of 1 over x squared open square brackets fraction numerator e to the power of x plus e to the power of negative x end exponent minus 2 over denominator x squared end fraction minus 1 close square brackets end exponent
U sin g space t h e space e x p a n s i o n space o f space e to the power of x space a n d space e to the power of negative x end exponent comma space e to the power of x plus e to the power of negative x end exponent equals 2 plus fraction numerator 2 x squared over denominator 2 factorial end fraction plus fraction numerator 2 x to the power of 4 over denominator 4 factorial end fraction plus... plus infinity
L equals limit as x rightwards arrow 0 of space e to the power of 1 over x squared open square brackets fraction numerator 2 plus begin display style fraction numerator 2 x squared over denominator 2 factorial end fraction end style plus begin display style fraction numerator 2 x to the power of 4 over denominator 4 factorial end fraction end style plus... plus infinity minus 2 over denominator x squared end fraction minus 1 close square brackets end exponent
equals limit as x rightwards arrow 0 of space e to the power of open square brackets fraction numerator begin display style fraction numerator 2 x squared over denominator 2 factorial end fraction end style plus begin display style fraction numerator 2 x to the power of 4 over denominator 4 factorial end fraction end style plus begin display style fraction numerator 2 x to the power of 6 over denominator 6 factorial end fraction end style plus... plus infinity over denominator x to the power of 4 end fraction minus 1 over x squared close square brackets end exponent
equals limit as x rightwards arrow 0 of space e to the power of open square brackets fraction numerator begin display style x squared plus x to the power of 4 over 12 plus... plus infinity minus x squared end style over denominator x to the power of 4 end fraction close square brackets end exponent
equals limit as x rightwards arrow 0 of space e to the power of open square brackets fraction numerator begin display style x to the power of 4 over 12 plus x to the power of 6 over 360... plus infinity end style over denominator x to the power of 4 end fraction close square brackets end exponent
equals limit as x rightwards arrow 0 of e to the power of open square brackets 1 over 12 plus x squared over 360 plus... infinity close square brackets end exponent
equals e to the power of 1 over 12 end exponent

Answered by Renu Varma | 1st Jun, 2020, 11:44: AM