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

for \(f(x)\) to be continuous at \(x=2\), we must hav \(\lim\limits_{x\to 2}f(x) = f(2)\)
that is :
\[\lim\limits_{x\to 2}\frac{\sqrt{2x+5}-\sqrt{x+7}}{x-2} = k\]

do that conjugate thingy and try evaluating the limit