To evaluate this limit, we can try to simplify the expression first.
Step 1: Rationalize the numerator
To eliminate the square roots in the numerator, we will rationalize it by multiplying both the numerator and denominator by the conjugate of the numerator, which is the expression obtained by changing the sign of each square root term.
\[
\begin{aligned} \lim _{x \rightarrow 2}\left(\frac{\sqrt{x+98}-\sqrt{x+14}-\sqrt{x+1}-\sqrt{x+2}-\sqrt{x-1}}{x-2}\right) &= \lim _{x \rightarrow 2}\left(\frac{\sqrt{x+98}-\sqrt{x+14}-\sqrt{x+1}-\sqrt{x+2}-\sqrt{x-1}}{x-2} \cdot \frac{\sqrt{x+98}+\sqrt{x+14}+\sqrt{x+1}+\sqrt{x+2}+\sqrt{x-1}}{\sqrt{x+98}+\sqrt{x+14}+\sqrt{x+1}+\sqrt{x+2}+\sqrt{x-1}}\right) \\ &= \lim _{x \rightarrow 2}\left(\frac{(\sqrt{x+98})^2 - (\sqrt{x+14})^2 - (\sqrt{x+1})^2 - (\sqrt{x+2})^2 - (\sqrt{x-1})^2}{(x-2)(\sqrt{x+98}+\sqrt{x+14}+\sqrt{x+1}+\sqrt{x+2}+\sqrt{x-1})}\right) \end{aligned}
\]
Notice that the numerators now contain squared terms, which will simplify the expression.
Step 2: Simplify the numerator
\[
\begin{aligned} \lim _{x \rightarrow 2}\left(\frac{(\sqrt{x+98})^2 - (\sqrt{x+14})^2 - (\sqrt{x+1})^2 - (\sqrt{x+2})^2 - (\sqrt{x-1})^2}{(x-2)(\sqrt{x+98}+\sqrt{x+14}+\sqrt{x+1}+\sqrt{x+2}+\sqrt{x-1})}\right) &= \lim _{x \rightarrow 2}\left(\frac{x+98 - x-14 - x-1 - x-2 - x+1}{(x-2)(\sqrt{x+98}+\sqrt{x+14}+\sqrt{x+1}+\sqrt{x+2}+\sqrt{x-1})}\right) \\ &= \lim _{x \rightarrow 2}\left(\frac{98 -14 - 1 - 2}{\sqrt{x+98}+\sqrt{x+14}+\sqrt{x+1}+\sqrt{x+2}+\sqrt{x-1}}\right) \\ &= \lim _{x \rightarrow 2}\left(\frac{81}{\sqrt{x+98}+\sqrt{x+14}+\sqrt{x+1}+\sqrt{x+2}+\sqrt{x-1}}\right) \end{aligned}
\]
Step 3: Evaluate the limit
Now we can evaluate the limit by substituting the value of x into the simplified expression:
\[
\lim _{x \rightarrow 2}\left(\frac{81}{\sqrt{x+98}+\sqrt{x+14}+\sqrt{x+1}+\sqrt{x+2}+\sqrt{x-1}}\right) = \frac{81}{\sqrt{100}+\sqrt{16}+\sqrt{3}+\sqrt{4}+\sqrt{1}} = \frac{81}{10+4+\sqrt{3}+2+1} = \frac{81}{17+\sqrt{3}}
\]
Therefore, the limit of the given expression as x approaches 2 is \(\frac{81}{17+\sqrt{3}}\).