1. **State the problem:** Find the following limits for the given function $f(x)$ based on the graph: - $\lim_{x \to -1^-} f(x)$ - $\lim_{x \to -1^+} f(x)$ - $\lim_{x \to 1} f(x)$ - $\lim_{x \to \infty} f(x)$ - $\lim_{x \to -\infty} f(x)$ 2. **Recall limit concepts:** - The left-hand limit $\lim_{x \to a^-} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the left. - The right-hand limit $\lim_{x \to a^+} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the right. - If both one-sided limits exist and are equal, the two-sided limit $\lim_{x \to a} f(x)$ exists and equals that value. - Vertical asymptotes occur where the function grows without bound (to $\infty$ or $-\infty$). - Horizontal asymptotes describe the behavior as $x \to \pm \infty$. 3. **Analyze each limit from the graph description:** - Near $x = -1$: - From the left ($x \to -1^-$), the function oscillates but approaches a vertical asymptote near $x = -1$. The graph suggests the function approaches $-\infty$ or $\infty$; the problem states oscillations approaching a vertical asymptote near $x = -1$. - From the right ($x \to -1^+$), the function rises steeply upward, so $f(x) \to \infty$. - Near $x = 1$: - The graph has a vertical asymptote near $x = 1$, so $\lim_{x \to 1} f(x)$ does not exist (function tends to $\infty$ or $-\infty$). - As $x \to \infty$: - The function approaches a horizontal line near $y = 1$, so $\lim_{x \to \infty} f(x) = 1$. - As $x \to -\infty$: - The function oscillates on the left side, but the problem states oscillations approaching a vertical asymptote near $x = -1$, so as $x \to -\infty$, the function likely oscillates without settling to a limit. We assume the limit does not exist. 4. **Summarize the limits:** $$\lim_{x \to -1^-} f(x) = \text{does not exist due to oscillations}$$ $$\lim_{x \to -1^+} f(x) = \infty$$ $$\lim_{x \to 1} f(x) = \text{does not exist due to vertical asymptote}$$ $$\lim_{x \to \infty} f(x) = 1$$ $$\lim_{x \to -\infty} f(x) = \text{does not exist due to oscillations}$$