Question

Evaluate the following limits.
${lim}_{x\to \infty }({\left((x+1)(x+2)(x+3)(x+4)\right)}^{\frac{1}{4}}-x)$

Answer 1

$\underset{x\to \infty }{lim}({\left[\right(x+1\left)\right(x+2\left)\right(x+3\left)\right(x+4\left)\right]}^{1/4}-x)$

$\underset{x\to \infty }{lim}(x{\left[(1+\frac{1}{x})(1+\frac{2}{x})(1+\frac{3}{x})(1+\frac{4}{x})\right]}^{\frac{1}{4}}-x)$

$\underset{x\to \infty }{lim}x[{\left[(1+\frac{1}{x})(1+\frac{2}{x})(1+\frac{3}{x})(1+\frac{4}{x})\right]}^{\frac{1}{4}}-1]$

$\underset{x\to \infty }{lim}x[{\left[(1+\frac{3}{x}+\frac{2}{x^2})(1+\frac{7}{x}+\frac{12}{x^2})\right]}^{\frac{1}{4}}-1]$

$\underset{x\to \infty }{lim}x[{(1+\frac{7}{x}+\frac{12}{x^2}+\frac{3}{x}+\frac{21}{x^2}+\frac{36}{x^3}+\frac{2}{x^2}+\frac{14}{x^3}+\frac{24}{x^4})}^{\frac{1}{4}}-1]$

$\underset{x\to \infty }{lim}x[{(1+\frac{10}{x}+\frac{35}{x^2}+\frac{50}{x^3}+\frac{24}{x^4})}^{\frac{1}{4}}-1]$

$(1+x{)}^n\simeq 1f_x^n$ when $x<<1$

$\underset{x\to \infty }{lim}x[1+\frac{10}{4x}+\frac{35}{4x^2}....1]$

$\underset{x\to \infty }{lim}\frac{10}{4}+\frac{35}{4x}+\frac{50}{4x^2}...$

$=\frac{10}{4}$