\[ \begin{split} &\lim_{x \to \infty} \frac{\sqrt{1+x}+3x}{\sqrt{1+x^2}}=\\[16pt] &=\lim_{x \to \infty} \frac{\sqrt{1+x}+3x}{\sqrt{1+x^2}}\cdot \frac{\frac{1}{x}}{\frac{1}{x}}=\\[16pt] &=\lim_{x \to \infty} \frac{\sqrt{\dfrac{1}{x^2}+\dfrac{x}{x^2}}+\dfrac{3x}{x}}{\sqrt{\dfrac{1}{x^2}+\dfrac{x^2}{x^2}}}=\\[16pt] &=\lim_{x \to \infty} \frac{\sqrt{\dfrac{1}{x^2}+\dfrac{1}{x}}+3}{\sqrt{\dfrac{1}{x^2}+1}}=\\[16pt] &=\frac{3}{\sqrt{1}}=3 \end{split} \]