Matematica - Siruri de numere reale. Limite de functii

Siruri de numere reale. Limite de functii - Intrebarea 57

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Să se calculeze $limit as x rightwards arrow 0 of open parentheses fraction numerator a to the power of ln x end exponent plus b to the power of ln x end exponent over denominator a plus b end fraction close parentheses to the power of fraction numerator 1 over denominator ln x minus 1 end fraction end exponent comma space a comma b element of straight real numbers subscript plus superscript asterisk times$ .

root index a b of a to the power of a b to the power of b end root

root index a b of open parentheses a b close parentheses to the power of a plus b end exponent end root

root index a plus b of a to the power of a b to the power of b end root

Raspunsul este corect.

Raspunsul este gresit.

Explicatie:

Răspuns corect: C

$limit as x rightwards arrow 0 of open parentheses fraction numerator a to the power of ln x end exponent plus b to the power of ln x end exponent over denominator a plus b end fraction close parentheses to the power of fraction numerator 1 over denominator ln x minus 1 end fraction end exponent equals limit as x rightwards arrow 0 of open parentheses 1 plus fraction numerator a to the power of ln x end exponent plus b to the power of ln x end exponent minus a minus b over denominator a plus b end fraction close parentheses to the power of fraction numerator 1 over denominator ln x minus 1 end fraction end exponent equals limit as x rightwards arrow 0 of open square brackets open parentheses 1 plus fraction numerator a to the power of ln x end exponent plus b to the power of ln x end exponent minus a minus b over denominator a plus b end fraction close parentheses to the power of fraction numerator a plus b over denominator a to the power of ln x end exponent plus b to the power of ln x end exponent minus a minus b end fraction end exponent close square brackets to the power of fraction numerator a to the power of ln x end exponent plus b to the power of ln x end exponent minus a minus b over denominator a plus b end fraction times fraction numerator 1 over denominator ln x minus 1 end fraction end exponent equals e to the power of fraction numerator 1 over denominator a plus b end fraction limit as x rightwards arrow 0 of open square brackets fraction numerator a open parentheses a to the power of ln x minus 1 end exponent minus 1 close parentheses over denominator ln x minus 1 end fraction plus fraction numerator b open parentheses b to the power of ln x minus 1 end exponent minus 1 close parentheses over denominator ln x minus 1 end fraction close square brackets end exponent equals e to the power of fraction numerator 1 over denominator a plus b end fraction ln a to the power of a b to the power of b end exponent equals root index a plus b of a to the power of a b to the power of b end root$