Calculus: Early Transcendentals (2nd Edition)

$\lim\limits_{x \to 0}{\frac{sin (7x)}{sin(3x)}}=\frac{7}{3}$
$\lim\limits_{x \to 0}{\frac{sin (7x)}{sin(3x)}}$ Multiply and divide by 7 and 3 $=\lim\limits_{x \to 0}{\frac{\frac{7sin (7x)}{7x}}{\frac{3sin(3x)}{3x}}}$ Let $t=7x$ in numerator and $u = 3x$ in denominator $=\frac{7}{3}\frac{\lim\limits_{t \to 0}{\frac{sin(7x)}{7x}}}{\lim\limits_{u \to 0}{\frac{sin(3x)}{3x}}}$ $=\frac{7}{3}\times\frac{1}{1}=\frac{7}{3}$