Question

Use the definition of continuity to determine whether f is continuous at a. f(x) = 5x4 - 9x3 + x - 7a = 7

Answer 1

We have
`f(7)=8918`. By the definition of continuity, we require that

`\displaystyle\lim_(x\to7)f(x)=f(7)`

So for any
`\varepsilon>0`, we want to find sufficient
`\delta` for which

`0<|x-7|<\delta\implies|f(x)-8918|<\varepsilon`

We have

`|5x^4-9x^3+x-7-8918|=|5 x^4 - 9 x^3 + x - 8925|=|x - 7||5 x^3 + 26 x^2 + 182 x + 1275|`

Suppose we let
`0<\delta\le1`. Then

`|x-7|\le1\implies x\le8`

so that

`|5 x^3 + 26 x^2 + 182 x + 1275|\le5|x|^3+26|x|^2+182|x|+1275\le6955`

`\implies|5x^4-9x^3+x-7-8918|\le6955|x-7|<\varepsilon`

`\implies|x-7|<\frac\varepsilon{6955}`

which suggests that it suffices to choose

`\delta=\min\left\{1,\frac\varepsilon{6955}\right\}`

in order to meet the required condition.