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)=(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\\e-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}`