To analyze this graph and answer the question, let’s evaluate the characteristics of $f(x)$ as indicated by the options given.

From the image, we observe that:

1. There is a vertical asymptote at $x = 2$, as the function heads toward infinity on both sides as $x$ approaches 2 from the left and right.
2. There appears to be a diagonal asymptote as $x \to \pm \infty$, with the graph approaching the line $y = -x + 2$.

Evaluating each option:

1. $\lim_{x \to 1} f(x) = 0$:

   • The graph doesn’t show any specific behavior at $x = 1$. It doesn’t appear that $f(x)$ approaches 0 as $x \to 1$, so this option is likely incorrect.

2. $\lim_{x \to 2^-} f(x) = +\infty$:

   • As $x$ approaches 2 from the left, $f(x)$ does go to $+\infty$, which makes this option correct.

3. $\lim_{x \to 2^+} f(x) = +\infty$:

   • As $x$ approaches 2 from the right, $f(x)$ also goes to $+\infty$, confirming this option is correct as well.

4. $\lim_{x \to +\infty} f(x) = +\infty$:

   • As $x \to +\infty$, the function does not tend toward $+\infty$; instead, it appears to approach the line $y = -x + 2$. So, this option is incorrect.

5. Asymptote $y = x + 2$:

   • The graph does not show the line $y = x + 2$ as an asymptote, so this is incorrect.

6. Asymptote $y = -x + 2$:

   • The line $y = -x + 2$ appears to be a slant asymptote, as the function seems to approach this line as $x \to \pm \infty$, so this option is correct.

Correct Answers:

Based on the analysis, the correct options are:

• $\lim_{x \to 2^-} f(x) = +\infty$
• $\lim_{x \to 2^+} f(x) = +\infty$
• The asymptote $y = -x + 2$

Would you like more details on asymptotic behavior or limits?

Related Questions:

1. What is the formal definition of a slant asymptote, and how is it identified?
2. How do vertical asymptotes affect the behavior of a function as it approaches specific values?
3. Why does a function have different limits from the left and right at certain points?
4. What is the significance of horizontal and slant asymptotes in describing end behavior?
5. How can we determine asymptotes from the algebraic form of a function?

Tip: To find slant asymptotes algebraically, consider dividing the function by $x$ for large values of $x$ and examining the behavior of the quotient.