$\text{a)}$  

$\underset{x\to 0}{\lim }\frac{x}{1-\sqrt{x+1}}=\underset{x\to 0}{\lim }\frac{x\left(1+\sqrt{x+1}\right)}{1^2-{\sqrt{x+1}}^2}=\underset{x\to 0}{\lim }\frac{x\left(1+\sqrt{x+1}\right)}{1-\left|x+1\right|}=$  

$=\underset{x\to 0}{\lim }\frac{x\left(1+\sqrt{x+1}\right)}{1-x-1}=\underset{x\to 0}{\lim }\frac{x\left(1+\sqrt{x+1}\right)}{-x}=$  

$=\underset{x\to 0}{\lim }-\left(1+\sqrt{x+1}\right)=-\left(1+\sqrt{1}\right)=-2$  

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$\text{b)}$  

$\underset{x\to 4}{\lim }\frac{x-4}{\sqrt{x}-2}=\underset{x\to 4}{\lim }\frac{\left(x-4\right)\left(\sqrt{x}+2\right)}{{\sqrt{x}}^2-2^2}=\underset{x\to 4}{\lim }\frac{\left(x-4\right)\left(\sqrt{x}+2\right)}{\left|x\right|-4}=$  

$=\underset{x\to 4}{\lim }\frac{\left(x-4\right)\left(\sqrt{x}+2\right)}{x-4}=\underset{x\to 4}{\lim }\left(\sqrt{x}+2\right)=\sqrt{4}+2=2+2=4$  

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$\text{c)}$  

$\underset{x\to -2}{\lim }\frac{\sqrt{x+3}-1}{2x+4}=\underset{x\to -2}{\lim }\frac{\sqrt{x+3}-1}{2x+4}\cdot \frac{\sqrt{x+3}+1}{\sqrt{x+3}+1}=$  

$=\underset{x\to -2}{\lim }\frac{{\sqrt{x+3}}^2-1^2}{\left(2x+4\right)\left(\sqrt{x+3}+1\right)}=\underset{x\to -2}{\lim }\frac{\left|x+3\right|-1}{2\left(x+2\right)\left(\sqrt{x+3}+1\right)}=$  

$=\underset{x\to -2}{\lim }\frac{x+2}{2\left(x+2\right)\left(\sqrt{x+3}+1\right)}=\underset{x\to -2}{\lim }\frac{1}{2\left(\sqrt{x+3}+1\right)}=\frac{1}{2\left(\sqrt{-2+3}+1\right)}=$  

$=\frac{1}{2\left(1+1\right)}=\frac{1}{2\cdot 2}=\frac{1}{4}$  

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$\text{d)}$  

$\underset{x\to -3}{\lim }\frac{1-\sqrt{x+4}}{x+3}=\underset{x\to -3}{\lim }\frac{1-\sqrt{x+4}}{x+3}\cdot \frac{1+\sqrt{x+4}}{1+\sqrt{x+4}}=$  

$=\underset{x\to -3}{\lim }\frac{1^2-{\sqrt{x+4}}^2}{\left(x+3\right)\left(1+\sqrt{x+4}\right)}=\underset{x\to -3}{\lim }\frac{1-\left|x+4\right|}{\left(x+3\right)\left(1+\sqrt{x+4}\right)}=$  

$=\underset{x\to -3}{\lim }\frac{1-x-4}{\left(x+3\right)\left(1+\sqrt{x+4}\right)}=\underset{x\to -3}{\lim }\frac{-\left(x+3\right)}{\left(x+3\right)\left(1+\sqrt{x+4}\right)}=$  

$=\underset{x\to -3}{\lim }\frac{-1}{1+\sqrt{x+4}}=\frac{-1}{1+\sqrt{-3+4}}=\frac{-1}{1+1}=-\frac{1}{2}$