Question

Find the value of the following limit:
$\underset{x\to \infty }{lim}\frac{(x-1)(x-2)(x-3)(x-4)(x-5)}{(5x-1{)}^5}$

Answer 1

$\underset{x\to \infty }{lim}\frac{(x-1)(x-2)(x-3)(x-4)(x-5)}{(5x-1{)}^5}$

$=\underset{x\to \infty }{lim}\frac{x(1-\frac{1}{x})\cdot x(1-\frac{2}{x})\cdot x(1-\frac{3}{x})\cdot x(1-\frac{4}{x})\cdot x(1-\frac{5}{x})}{x^5{(5-\frac{1}{x})}^5}$

$=\underset{x\to \infty }{lim}\frac{x^5(1-\frac{1}{x})(1-\frac{2}{x})(1-\frac{3}{x})(1-\frac{4}{x})(1-\frac{5}{x})}{x^5{(5-\frac{1}{x})}^5}$

$=\frac{(1-\frac{1}{\infty })(1-\frac{2}{\infty })(1-\frac{3}{\infty })(1-\frac{4}{\infty })(1-\frac{5}{\infty })}{{(5-\frac{1}{\infty })}^5}$

$=\frac{1\times 1\times 1\times 1\times 1}{5^5}=5^{-5}$.