Question

Check for the discontinuity of the function and tell if the discontinuity is removable or not

Answer 1

$\begin{gathered} \text{ To check what are the points of discontinuity of a rational function} \\ f(x)=\frac{g(x)}{h(x)} \\ \text{ we have to search values where h(x) = 0} \end{gathered}$$\begin{gathered} \text{ So we will search the value where } \\ x-7=0 \\ \text{ the value is } \\ x=7 \end{gathered}$$\begin{gathered} \text{ We will see if the discontinuity is removable. A discontinuity at a point p} \\ \text{ is removable if } \\ \lim _{x\to p^-}f(x)=\lim _{x\to p^+}f(x) \\ \text{ To see that, we will look at the graph of the function} \end{gathered}$$\begin{gathered} \text{From the graph we get } \\ \lim _{x\to7^-}f(x)=-1 \\ \text{but } \\ \lim _{x\to7^+}f(x)=1 \\ \text{ Since } \\ 1\ne-1 \\ \text{the discontinuity is not removable} \end{gathered}$$\begin{gathered} \text{ In case the discontinuity is removable, we modify the function by setting } \\ f(p)=\lim _{x\to p}f(x) \end{gathered}$