Find the limit.
\[
\lim _{x \rightarrow 1} \frac{\sqrt{x^{2}+15}-4}{x-1}
\]

Step 1 ：We are given the limit problem \(\lim _{x \rightarrow 1} \frac{\sqrt{x^{2}+15}-4}{x-1}\). This is a limit problem with an indeterminate form of type 0/0. We can use L'Hopital's Rule to solve it.

Step 2 ：L'Hopital's Rule states that if the limit of a function as x approaches a certain value is of the form 0/0 or ∞/∞, 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 ：So, we need to find the derivative of the numerator and the derivative of the denominator, and then find the limit of the ratio of these two derivatives as x approaches 1.

Step 4 ：The derivative of the numerator \(\sqrt{x^{2}+15}-4\) is \(\frac{x}{\sqrt{x^{2}+15}}\).

Step 5 ：The derivative of the denominator \(x-1\) is 1.

Step 6 ：Then, we find the limit of the ratio of these two derivatives as x approaches 1, which is \(\frac{1}{4}\).

Step 7 ：Final Answer: The limit of the function as x approaches 1 is \(\boxed{\frac{1}{4}}\).