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

Apr 22, 2018

Take ${\lim}_{x \to - {1}^{+}} f \left(x\right) , {\lim}_{x \to - {1}^{-}} f \left(x\right)$. If these limits are equal, we have continuity at $x = - 1.$

#### Explanation:

Take the left and right hand limits of $f \left(x\right)$ as $x \to - 1$. If

${\lim}_{x \to - {1}^{+}} f \left(x\right) = {\lim}_{x \to - {1}^{-}} f \left(x\right)$, then $f \left(x\right)$ is continuous at $x = - 1 :$

${\lim}_{x \to - {1}^{+}} {\left(x + 2 {x}^{3}\right)}^{4} = {\left(- 1 + 2 {\left(- 1\right)}^{3}\right)}^{4} = {\left(- 3\right)}^{4}$

${\lim}_{x \to - {1}^{-}} {\left(x + 2 {x}^{3}\right)}^{4} = {\left(- 1 + 2 {\left(- 1\right)}^{3}\right)}^{4} = {\left(- 3\right)}^{4}$

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, $f \left(x\right)$ is continuous at $x = - 1.$