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)?

1 Answer
Feb 9, 2017

Answer:

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.