1. **State the problem:** We need to find the limits of the function $f(x)$ approaching $x=-1$ from the left, from the right, and overall at $x=-1$. 2. **Find the left-hand limit $\lim_{x \to -1^-} f(x)$:** - From the graph description, as $x$ approaches $-1$ from the left, the function follows the straight line segment ending at $(-1,3)$ with a solid dot. - The solid dot at $x=-1$ means the value of the function there is $3$ from the left side. - Therefore, $$\lim_{x \to -1^-} f(x) = 3.$$ 3. **Find the right-hand limit $\lim_{x \to -1^+} f(x)$:** - As $x$ approaches $-1$ from the right, the graph shows a curve starting at an open circle at $(-1,1)$ (not included) and then continues onwards. - The value at $x=-1$ on the right side approaches 1 (open circle implies function is not defined there or does not equal 1). - Hence $$\lim_{x \to -1^+} f(x) = 1.$$ 4. **Find the overall limit $\lim_{x \to -1} f(x)$ if it exists:** - Since $$\lim_{x \to -1^-} f(x) = 3 \neq \lim_{x \to -1^+} f(x) = 1,$$ - The left and right limits are not equal. - Therefore, $$\lim_{x \to -1} f(x)$$ does not exist. - Explanation: For a limit at a point to exist, the left-hand and right-hand limits must be equal. Here, they differ, so the overall limit does not exist.