Chapter 14 - A Preview of Calculus: The Limit, Derivative, and Integral of a Function - 14.2 Algebra Techniques for Finding Limits - 14.2 Assess Your Understanding - Page 884: 55

$\frac{3}{4}.$

$$\lim_{x\to 0}\frac{3\sin{x}+\cos{x}-1}{4x}\\=\lim_{x\to 0}\frac{3\sin{x}}{4x}+\lim_{x\to 0}\frac{\cos{x}-1}{4x}\\=\frac{3}{4}(\lim_{x\to 0}\frac{\sin{x}}{x})+\frac{1}{4}(\lim_{x\to 0}\frac{\cos{x}-1}{x})\\=\frac{3}{4}(1)+\frac{1}{4}(0)\\=\frac{3}{4}.$$