10th Edition

Ron Larson + 1 other

ISBN: 9781285057095

Textbook Problem

The Gamma Function The Gamma Function Γ ( n ) is defined in terms of the integral of the function given by f(x) = xⁿ ^– ¹ e^–x, n > 0. Show that for any fixed value of n, the limit of f(x) as x approaches infinity is zero.

To determine

To prove: That for any fixed value of n, the limit of f(x) as x approaches infinity is zero.

The Gamma Function Γ(x) is defined in terms of the integral of the function given by:

f(x)=xn−1e−x, n>0.

Since, the Gamma Function is given by f(x)=xn−1e−x.

As the value of x approaches infinity, that is, x→∞. Solve limx→∞f(x).

limx→∞f(x)=limx→∞xn−1e−x=limx→∞xn−1ex=∞∞

As the value of limx→∞f(x) comes out to be ∞∞,

Therefore, L’Hopital rule can be applied. Apply L’Hopital rule.

limx→∞f(x)=(n−1)xn−2ex

On applying L’Hopital rule k−1 more times

Check out a sample textbook solution.

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Calculus

The Gamma Function The Gamma Function Γ ( n ) is defined in terms of the integral of the function given by f(x) = xn – 1 e–x, n > 0. Show that for any fixed value of n, the limit of f(x) as x approaches infinity is zero.

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Math
Calculus Calculus 10th Edition

The Gamma Function The Gamma Function Γ ( n ) is defined in terms of the integral of the function given by f(x) = xⁿ ^– ¹ e^–x, n > 0. Show that for any fixed value of n, the limit of f(x) as x approaches infinity is zero.