Find the limit. Use I'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. \[ \lim _{x \rightarrow 0} \frac{\tan (7 x)}{\sin (6 x)} \]

Step 1 ：We are given the limit \(\lim _{x \rightarrow 0} \frac{\tan (7 x)}{\sin (6 x)}\).

Step 2 ：As x approaches 0, both the numerator and the denominator approach 0. This is an indeterminate form, so we can apply L'Hopital's Rule.

Step 3 ：L'Hopital's Rule states that the limit of a quotient of two functions as x approaches a certain value is equal to the limit of the quotients of their derivatives, provided the limit exists.

Step 4 ：First, we find the derivative of the numerator, \(\tan(7x)\), which is \(7\sec^2(7x)\).

Step 5 ：Next, we find the derivative of the denominator, \(\sin(6x)\), which is \(6\cos(6x)\).

Step 6 ：We then find the limit of the quotient of these derivatives as x approaches 0, which simplifies to \(\frac{7}{6}\).

Step 7 ：Thus, the limit of the given function as x approaches 0 is \(\boxed{\frac{7}{6}}\).