Evaluate the limit \[ \lim _{x \rightarrow \infty} \frac{\ln \left(9+x^{2}\right)}{\ln \left(3+x^{7}\right)} \]

Step 1 ：Let's evaluate the limit \(\lim _{x \rightarrow \infty} \frac{\ln \left(9+x^{2}\right)}{\ln \left(3+x^{7}\right)}\)

Step 2 ：First, we differentiate the numerator and denominator separately. The derivative of the numerator, \(\ln \left(9+x^{2}\right)\), is \(\frac{2x}{x^{2} + 9}\). The derivative of the denominator, \(\ln \left(3+x^{7}\right)\), is \(\frac{7x^{6}}{x^{7} + 3}\)

Step 3 ：Next, we take the limit of the ratio of these two derivatives as \(x\) approaches infinity. This gives us \(\frac{2}{7}\)

Step 4 ：Final Answer: The limit of the given function as \(x\) approaches infinity is \(\boxed{\frac{2}{7}}\)