Monster!

Level 2

Evaluate this deadly limit

$\large \displaystyle \lim_{x \to 0^+ } \frac {2 \sin \sqrt{x^2 + \sqrt{x^3}} +\ln (1+x)}{x + \sqrt{x \sqrt{x}} }$

1 solution

Pi Han Goh
Dec 24, 2013

For small $x$ , $\sin(x) \approx x$ , and $\ln (1+x) \approx x$

$\large \frac { 2 \sqrt {x^2 + x^{3/2}} + x }{x + x^{3/4} } = \frac { 2 x^{3/4} \sqrt {x^{1/2} + 1} + x }{x + x^{3/4} }$

Divide numerator and denominator by $x^{3/4}$

$\large = \frac { 2 \sqrt {x^{1/2} + 1} + x^{1/4} }{x^{1/4} + 1 }$

Set $x$ approaches $0$ from the right

$\large = \frac { 2 \sqrt {0+1} + 0 }{0 + 1} = \boxed{2}$