<s>[INST] How do I make a change of variable for $\;\lim_{x\to 0}\frac{\sqrt{x^2+p^2}-p}{\sqrt{x^2+q^2}-q}$? I can't use l'hopital, so change of variable is the only way.
$$\lim_{x\to 0}\frac{\sqrt{x^2+p^2}-p}{\sqrt{x^2+q^2}-q}$$
 [/INST] We have: $\dfrac{\sqrt{x^2+p^2}-p}{\sqrt{x^2+q^2}-q}=\dfrac{\dfrac{x^2}{\sqrt{x^2+p^2}+p}}{\dfrac{x^2}{\sqrt{x^2+q^2}+q}}=\dfrac{\sqrt{x^2+q^2}+q}{\sqrt{x^2+p^2+p}}\to \dfrac{2q}{2p} = \dfrac{q}{p}$ as $x \to 0$.
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