Given the equation $f(x)= 1 + |x + 2|$ in x = -2, I want to evaluate the limit in x = -2, and the derivative of the function to look for continuity, and to check if its possible to get $f'$.
I don't know how to evaluate limits of abs, and then the derivative.
Thanks.
Consider what happens when you approach $x = -2$ from both sides. From the right-hand side, $x > -2$, so we have $f(x) = 1 + x + 2$. Thus, the limit as we approach from the right is $f(-2) = 1$. From the left-hand side, $ x < -2$, so $f(x) = 1 - (x+2)$, and still $f(-2) = 1$. Since absolute value is continuous, you can conclude that the limit as $x$ approaches $-2$ of $f$ is $1$.
Now consider the derivative of $f$. When we approach $x = -2$ from the right, $f' = 1$, yet when we approach from the left, $f' = -1$. At the specific point $x = -2$, the derivative is thus undefined. (Everywhere else it is defined piecewise.)