1. $\lim_{x \to 0^+} \sqrt{\dfrac{1}{x}+2} - \sqrt{\dfrac{1}{x}}$
2. $\lim_{x \to 0^+} \dfrac{\sqrt{2-x^2}}{x}$
3. $\lim_{x \to 0} x \sin (\dfrac{1}{x})$
Don't know how to solve. Can someone help?
First:
$\lim_{x \to 0^+} \dfrac{\dfrac{1}{x}+2-\dfrac{1}{x}}{\sqrt{\dfrac{1}{x}+2}+\sqrt{\dfrac{1 }{x}}}
= \lim_{x \to 0^+} \dfrac{2\sqrt{x}}{\sqrt{1+2x}+1}
= \lim_{x \to 0^+}\dfrac{0}{2}=0
$
Second:
$\lim_{x \to 0^+}\sqrt{\dfrac{2- x^2}{x^2}}$
Conclude that the limit does not exist, because I can't manipulate the expression any more?
Third:
Still confused.