<s>[INST] Calculate $ \lim_{x \to 4} \frac{3 -\sqrt{5 -x}}{1 -\sqrt{5 -x}} $ How evaluate the following limit?
$$ \lim_{x \to 4} \frac{3 -\sqrt{5 -x}}{1 -\sqrt{5 -x}} $$
I cannot apply L'Hopital because $ f(x) =  3 -\sqrt{5 -x} \neq 0 $ at $x = 5$
 [/INST] Let $\sqrt{5+x}=3+h$
Then $5+x=9+6h+h^2$
Let $\sqrt{5-x}=1+k$
Then $5-x=1+2k+k^2$
Adding gives: $5+x+5-x=9+6h+h^2+1+2k+k^2$
$0=6h+h^2+2k+k^2$
When $x=4$, $\sqrt{5+x}=\sqrt 9 =3 \Rightarrow 3+h=3 \Rightarrow h=0$
Similarly $k=0$
Required expression is $\frac {3-\sqrt{5+x}}{1-\sqrt{5-x}}=\frac h k$
$6h=-2k-k^2-h^2$
$\frac h k= -\frac 2 6 -\frac k6-\frac {h^2} {6k}$
As $h$ and $k$ tend to 0, $\frac h k$ tends to $-\frac 1 3$
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