Question

For what value \(k\) is the following function continuous at \(x=2\)?
\(f(x) = \begin{cases}
\frac{\sqrt{2x+5}-\sqrt{x+7}}{x-2} & x \neq 2 \\
k & x = 2
\end{cases}\)

Answer 1

You want to take the limit for the top part.

Answer 2

You can find the limit for the top part by multiplying top and bottom by the conjugate:\[
\sqrt{2x+5}+\sqrt{x+7}
\]