The value of k for which the function $f(x)=\left\{\begin{matrix}\frac{x^2+3x-10}{x-2} & x≠2\\k & x=2 \end{matrix}\right.$ is continuous at x=2; is :

7

The correct answer is Option (2) → 7

at $f(2)=k$

$\underset{x→2}{\lim}\frac{x^2+3x-10}{x-2}$

$x^2+3x-10=x^2+5x-2x-10$

$=x(x+5)-2(x+5)$

$=(x-2)(x+5)$

$⇒\underset{x→2}{\lim}\frac{(x-2)(x+5)}{(x-2)}$

$⇒\underset{x→2}{\lim}(x+5)=7$

so $k=7$ for continuity to exist