Use l'Hôpital's Rule to evaluate $\lim _{x \rightarrow 0} \frac{7-7 \cos x}{3 x^{2}}$. Then determine the limit using limit laws and commonly known limits.

Step 1 ：Given the limit \(\lim _{x \rightarrow 0} \frac{7-7 \cos x}{3 x^{2}}\), we can see that it is of the form \(\frac{0}{0}\) as \(x\) approaches 0. Therefore, we can apply l'Hôpital's Rule.

Step 2 ：l'Hôpital's Rule states that if the limit of a function is of the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then the limit of that function is equal to the limit of the derivative of the numerator divided by the derivative of the denominator.

Step 3 ：The derivative of \(7-7\cos x\) with respect to \(x\) is \(7\sin x\) and the derivative of \(3x^2\) with respect to \(x\) is \(6x\).

Step 4 ：Substituting these derivatives back into the limit, we get \(\lim _{x \rightarrow 0} \frac{7\sin x}{6x}\).

Step 5 ：We can also determine the limit using limit laws and commonly known limits. We know that \(\lim_{x \rightarrow 0} \cos x = 1\) and \(\lim_{x \rightarrow 0} x^2 = 0\). Using these known limits and the limit laws, we can simplify the original limit expression to \(\lim_{x \rightarrow 0} \frac{7(1 - \cos x)}{3x^2}\).

Step 6 ：Using the limit law \(\lim_{x \rightarrow a} [f(x) - g(x)] = \lim_{x \rightarrow a} f(x) - \lim_{x \rightarrow a} g(x)\), we can split the limit into two separate limits: \(\lim_{x \rightarrow 0} \frac{7}{3x^2} - \lim_{x \rightarrow 0} \frac{7\cos x}{3x^2}\).

Step 7 ：The first limit is \(\infty\) and the second limit is 0, so the overall limit is \(\infty - 0 = \infty\). However, this result contradicts the result we obtained using l'Hôpital's Rule.

Step 8 ：Let's try a different approach. We know that \(\lim_{x \rightarrow 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}\), so we can rewrite the original limit as \(\lim_{x \rightarrow 0} \frac{7(1 - \cos x)}{3x^2} = \frac{7}{3} \lim_{x \rightarrow 0} \frac{1 - \cos x}{x^2} = \frac{7}{3} \cdot \frac{1}{2} = \frac{7}{6}\).

Step 9 ：This result agrees with the result we obtained using l'Hôpital's Rule, so it seems that our second approach using the limit laws and commonly known limits is correct.

Step 10 ：Final Answer: \(\boxed{\frac{7}{6}}\)