Question

Apply the lmitsto given expression $\underset{x\to 0}{lim}({\left(\right(x+1\left)\right(x+2\left)\right(x+3\left)\right(x+4\left)\right)}^{\frac{1}{4}}-x)$.

Answer 1

${lim}_{x\to 0}({(x+1)(x+2)(x+3)(x+4)}^{\frac{1}{4}}-x)$

$=({(0+1)(0+2)(0+3)(0+4)}^{\frac{1}{4}}-0)$

$={24}^{\frac{1}{4}}$

$={\left(\sqrt{24}\right)}^{\frac{1}{2}}$

$=\sqrt{\left(2\sqrt{6}\right)}$