Question

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

Short answer

The limit of the function \(f(x) = x^{n-1}e^{-x}\) as \(x\) approaches infinity is zero for any fixed value of \(n\).

Step-by-step solution

- Break down the function

The given function can be considered as a product of two functions: \(f(x) = g(x)h(x)\), where \(g(x) = x^{n-1}\) and \(h(x) = e^{-x}\). As \(x\) tends to infinity, we need to investigate what happens to \(g(x)\) and \(h(x)\) separately.

- Evaluate the limit of \(g(x)\)

For any fixed value of \(n > 0\), \(\lim_{{x \to \infty}} g(x)= \lim_{{x \to \infty}} x^{n-1} = \infty\). This indicates that as \(x\) approaches infinity, \(g(x)\) also tends to infinity.

- Evaluate the limit of \(h(x)\)

Since the base of the exponential function in \(h(x)\) is \(e\) (a positive constant less than 1), when the exponent gets large and negative (which happens when \(x\) gets large), the function \(h(x)\) tends to zero. Mathematically, \(\lim_{{x \to \infty}} h(x) = \lim_{{x \to \infty}} e^{-x} = 0\). As \(x\) approaches infinity, \(h(x)\) diminishes to zero.

- Evaluate the limit of \(f(x)\)

Because \(f(x) = g(x)h(x)\), when \(x\) tends towards infinity, one part of the function (\(g(x)\)) tends towards infinity while the other part (\(h(x)\)) tends towards zero. We have to deal with an indeterminate form of type '0 times ∞'. We need to use L'Hopital's rule and rewrite the limit: \(\lim_{{x \to \infty}} f(x) = \lim_{{x \to \infty}} g(x)h(x) = \lim_{{x \to \infty}} \frac{g(x)}{\frac{1}{h(x)}}\). Now, we can compute the limit by taking the derivative of the numerator and the denominator. The derivative of \(g(x)\) is \((n-1)x^{n-2}\) and the derivative of \(\frac{1}{h(x)}\) is \(e^x\). So, we have \(\lim_{{x \to \infty}} f(x) = \lim_{{x \to \infty}} \frac{(n-1)x^{n-2}}{e^x}\). This is still an indeterminate form of type '0 times ∞'. Repeat the process again. If you continue this process until the numerator is a constant, every term in the numerator will be eliminated by the exponential in the denominator and the limit will be zero.