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The value of the limit x tend to infinity gamma (x+1)/(x/e)^x√x is A. √2π B. 1/2 C. 1/2π D. √π/2?
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The value of the limit x tend to infinity gamma (x+1)/(x/e)^x√x is A. ...
- **Limit Calculation**
To find the value of the limit as x tends to infinity of gamma(x+1)/(x/e)^x√x, we can use the properties of limits and the Stirling's approximation for the gamma function.
- **Stirling's Approximation**
Stirling's approximation states that for large x, gamma(x+1) is approximately equal to √(2πx)(x/e)^x.
- **Applying Stirling's Approximation**
Substitute the Stirling's approximation into the original limit expression:
= (√(2πx)(x/e)^x) / (x/e)^x√x
= √(2πx) / e^(√x)
- **Simplifying the Expression**
As x tends to infinity, the expression simplifies to:
= √(2π*∞) / e^∞
= √(2π) / ∞
= 0
Therefore, the value of the limit as x tends to infinity of gamma(x+1)/(x/e)^x√x is 0.
To find the value of the limit as x tends to infinity of gamma(x+1)/(x/e)^x√x, we can use the properties of limits and the Stirling's approximation for the gamma function.
- **Stirling's Approximation**
Stirling's approximation states that for large x, gamma(x+1) is approximately equal to √(2πx)(x/e)^x.
- **Applying Stirling's Approximation**
Substitute the Stirling's approximation into the original limit expression:
= (√(2πx)(x/e)^x) / (x/e)^x√x
= √(2πx) / e^(√x)
- **Simplifying the Expression**
As x tends to infinity, the expression simplifies to:
= √(2π*∞) / e^∞
= √(2π) / ∞
= 0
Therefore, the value of the limit as x tends to infinity of gamma(x+1)/(x/e)^x√x is 0.
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The value of the limit x tend to infinity gamma (x+1)/(x/e)^x√x is A. √2π B. 1/2 C. 1/2π D. √π/2?
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The value of the limit x tend to infinity 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 The value of the limit x tend to infinity 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 The value of the limit x tend to infinity gamma (x+1)/(x/e)^x√x is A. √2π B. 1/2 C. 1/2π D. √π/2?.
The value of the limit x tend to infinity 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 The value of the limit x tend to infinity 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 The value of the limit x tend to infinity gamma (x+1)/(x/e)^x√x is A. √2π B. 1/2 C. 1/2π D. √π/2?.
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