# Thread: A few limit problems

1. ## A few limit problems

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?

2. ## Re: A few limit problems

For the first use $\displaystyle (a- b)\frac{a+ b}{a+ b}= \frac{a^2- b^2}{a+ b}$. Then multiply both numerator and denominator by $\displaystyle \sqrt{x}$.

For the second, write it as $\displaystyle \sqrt{\frac{2- x^2}{x^2}}$

For the third, $\displaystyle -1\le sin(1/x)\le 1$.

3. ## Re: A few limit problems

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.

4. ## Re: A few limit problems

Also, regarding the second problem. Is it DNE or undefined? The answer in my book was DNE, but I don't see why it can't be just undefined. What's the difference?