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Lim x tend to gamma (x+1)/(x/e)^x√x is A. √2π. B. 1/2. C. 1/2π. D. √π/2.?
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Lim x tend to gamma (x+1)/(x/e)^x√x is A. √2π. B. 1/2. C. 1/2π. D. √π/...
Approach:
To find the limit of the given function as x approaches gamma, we can use L'Hopital's Rule to simplify the expression and evaluate the limit.
Given Function:
\[ \lim_{x \to \gamma} \frac{x+1}{(x/e)^x \sqrt{x}} \]
Apply L'Hopital's Rule:
\[ \lim_{x \to \gamma} \frac{x+1}{(x/e)^x \sqrt{x}} = \lim_{x \to \gamma} \frac{1}{(1/e)^x \sqrt{x} + x(e/x)^x(1/2x^{-1/2})} \]
\[ = \lim_{x \to \gamma} \frac{1}{e^{-x} \sqrt{x} + e^x/2} \]
Substitute x = gamma:
\[ = \frac{1}{e^{-\gamma} \sqrt{\gamma} + e^\gamma/2} \]
Final Answer:
\[ = \frac{1}{\sqrt{\gamma}e^{-\gamma} + e^\gamma/2} \]
Therefore, the limit of the given function as x approaches gamma is the above expression. The answer does not simplify further to the provided options A, B, C, or D.
To find the limit of the given function as x approaches gamma, we can use L'Hopital's Rule to simplify the expression and evaluate the limit.
Given Function:
\[ \lim_{x \to \gamma} \frac{x+1}{(x/e)^x \sqrt{x}} \]
Apply L'Hopital's Rule:
\[ \lim_{x \to \gamma} \frac{x+1}{(x/e)^x \sqrt{x}} = \lim_{x \to \gamma} \frac{1}{(1/e)^x \sqrt{x} + x(e/x)^x(1/2x^{-1/2})} \]
\[ = \lim_{x \to \gamma} \frac{1}{e^{-x} \sqrt{x} + e^x/2} \]
Substitute x = gamma:
\[ = \frac{1}{e^{-\gamma} \sqrt{\gamma} + e^\gamma/2} \]
Final Answer:
\[ = \frac{1}{\sqrt{\gamma}e^{-\gamma} + e^\gamma/2} \]
Therefore, the limit of the given function as x approaches gamma is the above expression. The answer does not simplify further to the provided options A, B, C, or D.
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Lim x tend to gamma (x+1)/(x/e)^x√x is A. √2π. B. 1/2. C. 1/2π. D. √π/2.?
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Lim x tend to gamma (x+1)/(x/e)^x√x is A. √2π. B. 1/2. C. 1/2π. D. √π/2.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Lim x tend to gamma (x+1)/(x/e)^x√x is A. √2π. B. 1/2. C. 1/2π. D. √π/2.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Lim x tend to gamma (x+1)/(x/e)^x√x is A. √2π. B. 1/2. C. 1/2π. D. √π/2.?.
Lim x tend to gamma (x+1)/(x/e)^x√x is A. √2π. B. 1/2. C. 1/2π. D. √π/2.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Lim x tend to gamma (x+1)/(x/e)^x√x is A. √2π. B. 1/2. C. 1/2π. D. √π/2.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Lim x tend to gamma (x+1)/(x/e)^x√x is A. √2π. B. 1/2. C. 1/2π. D. √π/2.?.
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