4. $[0 / 1$ Points] DETAILS PREVIOUS ANSWERS SCALC9 3.4.008. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Evaluate the limit using the appropriate properties of limits. (If the limit is infinite, enter ' $\infty$ ' or '- $\infty$ ', as appropriate. If the limit does not otherwise exist, enter DNE.) \[ \lim _{x \rightarrow \infty}\left(\sqrt{\frac{49 x^{3}+5 x-9}{4-6 x+x^{3}}}\right) \] Need Help? Read It Submit Answer
Solution
Step 1 ：The given function is \(f = \sqrt{\frac{49x^{3} + 5x - 9}{x^{3} - 6x + 4}}\).
Step 2 ：We can rewrite this function as \(f = \left(\frac{49x^{3} + 5x - 9}{x^{3} - 6x + 4}\right)^{0.5}\) to make it easier to differentiate.
Step 3 ：Next, we find the derivatives of the numerator and the denominator. The derivative of the numerator is \(147x^{2} + 5\) and the derivative of the denominator is \(3x^{2} - 6\).
Step 4 ：Since the limit is of the form \(\frac{\infty}{\infty}\), we can apply L'Hopital's rule. 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.
Step 5 ：Applying L'Hopital's rule, we find that the limit of the function as x approaches infinity is 49.
Step 6 ：Thus, the final answer is \(\boxed{49}\).
