Math Snap

STEP 1

Assumptions1. The function $f(x)$ is defined as $|x+|$ for all $x \neq -1$ and $-1$ for $x = -1$.
. The value of $c$ is $-1$.
3. We are checking for continuity at $x = c$.

STEP 2

A function $f(x)$ is continuous at a point $x = c$ if the following three conditions are met1. $f(c)$ is defined.
2. $\lim_{x \to c} f(x)$ exists.
. $\lim_{x \to c} f(x) = f(c)$.
We will check each of these conditions.

STEP 3

First, we check if $f(c)$ is defined. From the definition of the function, we know that $f(-1) = -1$.

STEP 4

Next, we need to calculate the limit of $f(x)$ as $x$ approaches $c$ from both the left and the right. If both of these limits exist and are equal, then $\lim_{x \to c} f(x)$ exists.

STEP 5

We calculate the left-hand limit, which is $\lim_{x \to c^-} f(x)$.
$$\lim_{x \to -1^-} f(x) = \lim_{x \to -1^-} |x+2|$$

STEP 6

Since $x$ is approaching $-1$ from the left (i.e., $x < -1$), the expression inside the absolute value, $x+2$, is negative. Therefore, $|x+2| = -(x+2)$.
$$\lim_{x \to -1^-} f(x) = \lim_{x \to -1^-} -(x+2)$$

STEP 7

Now, we substitute $x = -1$ into the expression to calculate the left-hand limit.
$$\lim_{x \to -1^-} f(x) = -(-1+2) = -1$$

STEP 8

Next, we calculate the right-hand limit, which is $\lim_{x \to c^+} f(x)$.
$$\lim_{x \to -1^+} f(x) = \lim_{x \to -1^+} |x+2|$$

STEP 9

Since $x$ is approaching $-$ from the right (i.e., $x > -$), the expression inside the absolute value, $x+2$, is positive. Therefore, $|x+2| = x+2$.
$$\lim_{x \to -^+} f(x) = \lim_{x \to -^+} (x+2)$$

STEP 10

Now, we substitute $x = -$ into the expression to calculate the right-hand limit.
$$\lim_{x \to -^+} f(x) = -+2 =$$

STEP 11

Since the left-hand limit ($-$) and the right-hand limit ($$) are not equal, $\lim_{x \to c} f(x)$ does not exist.

Therefore, the function $f(x)$ is not continuous at $x = c = -$, because it does not meet all three conditions for continuity at that point.