Let a > 0 be a real number. Then the limit \lim_{x\to 2} {a ^x +a^{3-x} - (a ^2+ a)} /{ a^{3-x }- a^{x\/2} } is -

KVPY SX SB Exam 2015 Question Paper

Question : 85 of 120

Marks: +1, -0

Let a > 0 be a real number. Then the limit

\lim↙{x→ 2} {a ^x +a^{3-x} - (a ^2+ a)} /{ a^{3-x }- a^{x\/2} }

is -

Solution:

\lim↙{x→ 2} {a ^x +a^{3-x} - (a ^2+ a)} /{ a^{3-x }- a^{x\/2} }(0/0)

Apply L hospital rule

\lim↙{x→ 2} {a ^xlna -a^{3-x}lna } /{ -a^{3-x }lna-1/2 a^{x\/2} lna}

{a ^2lna -a lna } /{ -a lna-a/2 a lna} ={a^2-a}/{{3a}/2}

=2/3(1-a)

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