Let e denote the base of the natural logarithm. The value of the real number a for which the right-hand limit lim x → 0 + 1 − x 1 χ − e − 1 x a is equal to a non-zero real number, is

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Let e denote the base of the natural logarithm. The value of the real number a for which the right-hand limit lim x &#8594; 0 + 1 &#8722; x 1 &#967; &#8722; e &#8722; 1 x a is equal to a non-zero real number, is

Quantrex Academy · Accepted Answer

Correct answer: A. A

lim x &#8594; 0 + 1 &#8722; x 1 x a x a &#8722; 1 d d x ln 1 &#8722; x x = lim x &#8594; 0 + 1 &#8722; x 1 x a x a &#8722; 1 1 x x &#8722; 1 &#8722; ln 1 &#8722; x x 2 = lim x &#8594; 0 + 1 &#8722; x 1 x a x a &#8722; 1 x &#8722; x &#8722; 1 ln 1 &#8722; x x 2 x &#8722; 1 = lim x &#8594; 0 + 1 &#8722; x 1 x x + x &#8722; 1 x + x 2 2 + x 3 3 . . . . . . . . &#8722; a 1 &#8722; x x a + 1 = lim x &#8594; 0 + 1 &#8722; x 1 x x 2 2 + x 3 1 2 &#8722; 1 3 + x 4 1 3 &#8722; 1 4 + &#8230; &#8722; a 1 &#8722; x x a + 1 Which is real number if and only if  a + 1 = 2 &#8658; a = 1

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