Question

Is the graph of `f (x) = (X^2 + x)/x` continuous on the interval [-4, 4]?

Answer 1

see below

Explanation:

To show that the function `f(x)=(x^2+x)/x` is continuous on [-4,4] we need to do the following:

1. Show that `lim_(x->-4^+) (x^2+x)/x =f(-4)`

2. Show that `lim_(x->4^-) (x^2+x)/x =f(4)`

That is,

1. `lim_(x->-4^+) (x^2+x)/x =lim_(x->-4^+) (x(x+1))/x`

`=lim_(x->-4^+)(cancelx(x+1))/cancelx=lim_(x->-4^+) x+1 = -3`

`f(-4)=(16-4)/-4 =12/-4 = -3`

Since `lim_(x->-4^+) (x^2+x)/x =f(-4)=-3`, f is continuous from the right at x = - 4.

2.`lim_(x->4^-) (x^2+x)/x = lim_(x->4^-) (x(x+1))/x`

`=lim_(x->4^-)(cancelx(x+1))/cancelx=lim_(x->4^-) x+1 = 5`

`f(4)=(16+4)/4=20/4=5`

Since `lim_(x->4^-) (x^2+x)/x =f(4)`, f is continuous from the left at 4.

Since f is continuous from the left at x = 4and continuous from the right at x = -4, hence f is continuous on the interval [-4,4].