Question

step 3 of 3: find \\(\lim_{x \to -1} f(x)\\).

Explanation:

Analyze the behavior of \(f(x)\) as \(x\) approaches \(-1\) from the left

As \(x \to -1^-\), we follow the curve from the left side toward \(x = -1\). The graph shows a series of oscillations. Specifically, at \(x = -1\), there is an open circle at the point \((-1, -2)\). Therefore, as \(x\) approaches \(-1\) from the left, the function values \(f(x)\) approach \(-2\):

$$ \lim_{x \to -1^-} f(x) = -2 $$

Analyze the behavior of \(f(x)\) as \(x\) approaches \(-1\) from the right

As \(x \to -1^+\), we follow the curve from the right side toward \(x = -1\). The graph shows oscillations continuing from the right, leading directly to the same open circle at \((-1, -2)\). Therefore, as \(x\) approaches \(-1\) from the right, the function values \(f(x)\) also approach \(-2\):

$$ \lim_{x \to -1^+} f(x) = -2 $$

Determine the two-sided limit

Since the left-hand limit and the right-hand limit are equal:

$$ \lim_{x \to -1^-} f(x) = \lim_{x \to -1^+} f(x) = -2 $$

The two-sided limit exists and is equal to this common value:

$$ \lim_{x \to -1} f(x) = -2 $$

Answer:

$$\lim_{x \to -1} f(x) = -2$$