The graph provided depicts a function $g(x)$ with a vertical asymptote at $x = -2$ (indicated by the red line). The curve suggests the following behaviors as $x$ approaches different values:

1. As $x$ approaches negative infinity ($x \to -\infty$):

   • The function appears to decrease without bound, suggesting $g(x)$ tends towards negative infinity as $x \to -\infty$.
   • Thus, $\lim_{x \to -\infty} g(x) = -\infty$.

2. As $x$ approaches positive infinity ($x \to +\infty$):

   • The function continues to rise without bound, indicating that $g(x)$ approaches positive infinity.
   • Thus, $\lim_{x \to +\infty} g(x) = +\infty$.

3. As $x$ approaches -2 from the right ($x \to -2^+$):

   • The function rises sharply, indicating that $g(x)$ approaches positive infinity.
   • This is consistent with the provided answer, $\lim_{x \to -2^+} g(x) = +\infty$.

4. As $x$ approaches -2 from the left ($x \to -2^-$):

   • The function falls rapidly, indicating that $g(x)$ approaches negative infinity.
   • Therefore, $\lim_{x \to -2^-} g(x) = -\infty$.

Filling in the Blanks:

1. $\lim_{x \to -\infty} g(f(x)) = -\infty$
2. $\lim_{x \to +\infty} g(f(x)) = +\infty$
3. $\lim_{x \to 1^+} g(f(x)) = +\infty$ (This is already correct.)
4. $\lim_{x \to 0} g(f(x))$: Since $x$ is not close to the asymptote $x = -2$, we would need more context or information about $f(x)$ or the specific behavior of $g(x)$ near $x = 0$. This limit might involve further information about the function $f(x)$ to make an accurate determination.

Would you like to explore more details on this or have any specific questions?

Relative Questions:

1. How do vertical asymptotes affect the limits of functions?
2. What happens to the limit of a function when approaching infinity?
3. How can you determine the end behavior of a function from its graph?
4. How do you find the limit of a composite function like $g(f(x))$?
5. What methods can be used to find limits involving infinity?

Tip: Always check the behavior of a function near its asymptotes to correctly determine the limits as the function approaches these critical points.