The question asks for the correct asymptotic values (limits) of several functions as $x$ approaches a certain value, likely infinity. I'll break down each expression to find the limits.

1. $y = \sqrt[3]{x}(x-1)^2$

As $x \to \infty$:

• $\sqrt[3]{x} \to \infty$
• $(x - 1)^2$ behaves like $x^2$, so this term also grows without bound.

Thus, $y \to \infty$. This function does not approach a finite value, so it is not relevant to the multiple-choice answers.

2. $y = \frac{1}{e^x} - x$

As $x \to \infty$:

• $\frac{1}{e^x} \to 0$ (exponentially decays)
• $-x \to -\infty$

Thus, $y \to -\infty$, which does not match any of the options provided.

3. $y = e^{x^2} + \frac{1}{1 + x^2} + 1$

As $x \to \infty$:

• $e^{x^2} \to \infty$ because the exponential growth of $e^{x^2}$ dominates.
• $\frac{1}{1 + x^2} \to 0$

So, $y \to \infty$, which again does not match the multiple-choice answers.

4. $y = \arctan \left( \frac{x^2 + 1}{x - 2} \right) + \frac{1}{x}$

As $x \to \infty$:

• $\frac{x^2 + 1}{x - 2} \approx \frac{x^2}{x} = x$ (for large $x$, the dominant term is $x$)
• $\arctan(x) \to \frac{\pi}{2}$
• $\frac{1}{x} \to 0$

Thus, $y \to \frac{\pi}{2} \approx 1.57$, which is close to 1.

Conclusion:

The answer should be b. 1 for the fourth function.

Would you like to discuss any of the details further, or need clarification?

Here are 5 follow-up questions based on this:

1. How does one calculate limits at infinity for different types of functions?
2. What are common techniques for evaluating limits involving exponential growth?
3. How does the behavior of $e^{x^2}$ compare with other rapidly growing functions?
4. What is the significance of the arctangent function as $x \to \infty$?
5. How can approximations help in simplifying limit calculations?

Tip: When evaluating limits, always check if any terms grow much faster than others. The term that dominates the growth usually dictates the behavior at infinity.