Problem

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.

Solution

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: \(\boxed{\frac{7}{6}}\)

From Solvely APP
Source: https://solvelyapp.com/problems/23929/

Get free Solvely APP to solve your own problems!

solvely Solvely
Download