Let $f(x)=\left\{\begin{array}{cl}\frac{x^{3}+x^{2}-16 x+20}{(x-2)^{2}}, & ext { if } x
eq 2 \

Question

Let $f(x)=\left\{\begin{array}{cl}\frac{x^{3}+x^{2}-16 x+20}{(x-2)^{2}}, & 	ext { if } x 
eq 2 \ k, & 	ext { if } x=2\end{array}ight.$ If $f$ is continuous for all $x$, then $k=$

Professor PC Thomas Classes · Answer

$f(x)=\frac{(x-2)^{2}(x+5)}{(x-2)^{2}}=x+5(x 
eq 2)$ $\lim _{x ightarrow 2} f(x)=2+5=7$ Since $f$ is continuous at $x=2, \lim _{x ightarrow 2} f(x)$\ $=f(2)=k$ $=f(2)=k$\ $	herefore \quad k=7$.\