Question

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \binom{\frac{x^3 + x^2 - 16x + 20}{\left( x - 2 \right)^2}, x \neq 2}{k, x = 2}\] 

 

Answer 1

Given:

\[f\left( x \right) = \binom{\frac{x^3 + x^2 - 16x + 20}{\left( x - 2 \right)^2}, x \neq 2}{k, x = 2}\]

\[\Rightarrow f\left( x \right) = \binom{\frac{x^3 + x^2 - 16x + 20}{x^2 - 4x + 4}, x \neq 2}{k, x = 2}\]

\[\Rightarrow f\left( x \right) = \binom{x + 5, x \neq 2}{k, x = 2}\]

If f(x) is continuous at x = 2, then

\[\lim_{x \to 2} f\left( x \right) = f\left( 2 \right)\]
\[ \Rightarrow \lim_{x \to 2} \left( x + 5 \right) = k\]
\[ \Rightarrow k = 2 + 5 = 7\]