limit $ \lim_{x \to 0} \frac{\sqrt{x}}{\sqrt{x+1} + \sqrt{x}} $

Question

$\lim_{x \to 0} \frac{\sqrt{x}}{\sqrt{x+1} + \sqrt{x}}$

Answer 1

Hint: If $$\lim_{x \rightarrow c} f(x)=l \;\;\;\text{and}\; \;\;\lim_{x \rightarrow c} g(x)=h \neq 0$$ then $$\lim_{x \rightarrow c} \frac{f(x)}{g(x)}=\frac{l}{h}$$

Answer 2

Recall that the first attempt to do when dealing with a limit is to check whether $f(x)$ is continuous at the finite limit point indeed in that case we have simply

$$\lim_{x\to x_0} f(x)=f(x_0)$$

otherwise whe need some other trick to determine whether ot not the limit exists and to determine it.

As already noticed in that case the function you are considering is continuous at $x=0$ and therefore the value for the limit is simply $f(0)$.

More interesting would be to evaluate

$$\lim_{x \to \infty} \frac{\sqrt{x}}{\sqrt{x+1} + \sqrt{x}}$$