[tex]\lim_{x\rightarrow 0}\frac{\sqrt{2019x+1}+(x-1)^{2019}}{sin(2019x)}=\\\lim_{x\rightarrow 0}\frac{\frac{\sqrt{2019x+1}+(x-1)^{2019}}{2019x}}{\frac{sin(2019x)}{2019x}}\\=\lim_{x\rightarrow 0}\frac{\sqrt{2019x+1}+(x-1)^{2019}}{2019x}\\=\lim_{x\rightarrow 0}\frac{\sqrt{2019x+1}-1+x^{2019}-C^1_{2019}x^{2018}+C^2_{2019}x^{2017}+...+C^{2018}_{2019}x}{2019x}\\=\lim_{x \to 0}\frac{\frac{2019x+1-1}{\sqrt{2019x+1}+1}}{2019x}+\lim_{x \to 0}\frac{x(x^{2018}-C^1_{2019}x^{2017}+C^2_{2019}x^{2016}+...+C^{2018}_{2019})}{2019x}\\=\lim_{x \to 0}\frac{1}{\sqrt{2019x+1}+1}+\lim_{x \to 0}\frac{C^{2018}_{2019}}{2019}=\frac{1}{2}+1=\frac{3}{2}[/tex]
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