Question

How do you prove that the function `f(x) = (x + 2x^3)^4` is continuous at a =-1?

Answer 1

Take `lim_(x->-1^+)f(x), lim_(x->-1^-)f(x)`. If these limits are equal, we have continuity at `x=-1.`

Explanation:

Take the left and right hand limits of `f(x)` as `x->-1`. If

`lim_(x->-1^+)f(x)=lim_(x->-1^-)f(x)`, then `f(x)` is continuous at `x=-1:`

`lim_(x->-1^+)(x+2x^3)^4=(-1+2(-1)^3)^4=(-3)^4`

`lim_(x->-1^-)(x+2x^3)^4=(-1+2(-1)^3)^4=(-3)^4`

Note that the direction from which we approached `-1` did not change how the limits were evaluated, as this is a polynomial.

These limits are equal; therefore, `f(x)` is continuous at `x=-1.`