Question

How do you use the definition of continuity and the properties of limits to show the function is continuous `F(x)= (x^2-8)^8` on the interval (-inf, inf)?

Answer 1

see below

Explanation:

Let a be any number from the interval `(-oo,oo)` then we need to show that `f(a)=lim_(x->a) f(x)`.

Since `f(x)=(x^2-8)^8` then

`f(a)=(a^2 -8)^8`

`lim_(x->a) (x^2-8)^8 =(lim_(x->a)x^2-lim_(x->a)8)^8 =(a^2-8)^8`

Since `f(a)=lim_(x->a) f(x)=(a^2-8)^8`,f is continuous at x=a for every a in `(-oo,oo)`

OR

We can look at `f(x)=(x^2-8)^8` and recognize that it is a polynomial and since polynomials are continuous everywhere therefore `f(x)=(x^2-8)^8` is continuous everywhere.