Question

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point;  \[f\left( x \right) = \begin{cases}\frac{x^2 - 25}{x - 5}, & x \neq 5 \\ k , & x = 5\end{cases}\]at x = 5

Answer 1

\[f\left( x \right) = \binom{\frac{x^2 - 25}{x - 5}, x \neq 5}{k, x = 5}\]

\[\Rightarrow f\left( x \right) = \binom{\frac{\left( x - 5 \right)\left( x + 5 \right)}{x - 5}, x \neq 5}{k, x = 5}\]

\[\Rightarrow f\left( x \right) = \binom{x + 5, x \neq 5}{k, x = 5}\]

If f(x) is continuous at x = 5, then

\[\lim_{x \to 5} f\left( x \right) = f\left( 5 \right)\]
\[ \Rightarrow \lim_{x \to 5} \left( x + 5 \right) = k\]
\[ \Rightarrow k = 5 + 5 = 10\]