Question

Find the value of k for which the function f (x ) =  \[\binom{\frac{x^2 + 3x - 10}{x - 2}, x \neq 2}{ k , x^2 }\] is continuous at x = 2 .

 

 

Answer 1

Given,

f (x ) =  \[\binom{\frac{x^2 + 3x - 10}{x - 2}, x \neq 2}{ k , x=2 }\]

\[\lim_{x \to 2^-} \left( \frac{x^2 + 3x - 10}{x - 2} \right) = \lim_{x \to 2^-} \left( x + 5 \right) = 7\]

\[f\left( 2 \right) = k\]

\[ \lim_{x \to 2^+} \left( \frac{x^2 + 3x - 10}{x - 2} \right) = \lim_{x \to 2^+} \left( x + 5 \right) = 7\]

If  f ( x)  is continuous at x  = 2 , then

\[\lim_{x \to 2^-} f(x) = f(2) = \lim_{x \to 2^+} f(x)\]

\[\Rightarrow k = 7\]