Question

If $f\left(x\right)=\frac{x^3+x^2-16x+20}{(x-2{)}^2},x\ne 2$

$=k,x=2$ is continuous at x=2, find the value of k.

Answer 1

The given function is continuous at $x=2$.

Therefore,

$li{m}_{x\to 2^{-}}f\left(x\right)=li{m}_{x\to 2^{+}}f\left(x\right)=f\left(2\right)$

$li{m}_{x\to 2}=f\left(2\right)$

Now, consider $li{m}_{x\to 2}f\left(x\right)=li{m}_{x\to 2}\frac{x^3+x^2-16x+20}{(x-2{)}^2}$

$=li{m}_{x\to 2}\frac{(x+5)(x-2{)}^2}{(x-2{)}^2}$

$=li{m}_{x\to 2}(x+5)=7$

Therefore,

$7=f\left(2\right)=k$

Hence, the required value of k is 7.