The image shows a set of problems related to checking asymptotic shifts for given functions. Here's a breakdown of the tasks and simpler explanations:

Problem Explanation:

You need to find the asymptotic shifts of functions $f(x)$, $g(x)$, and $h(x)$ in different combinations as $x \to \infty$ (as $x$ approaches infinity). Specifically, you're asked to evaluate the limit expressions:

1. $\lim_{x \to \infty} (f(x) + 2g(x))$
2. $\lim_{x \to \infty} (f(x) - h(x))$
3. $\lim_{x \to \infty} \left(\frac{f(x)}{g(x)}\right)$
4. $\lim_{x \to \infty} \left(\frac{g(x)}{f(x)}\right)$
5. $\lim_{x \to \infty} \left(\frac{h(x)}{f(x)}\right)$

Problems Given:

1. (59):

   • $f(x) = x^2$
   • $g(x) = x^2 - 3x - 1$

2. (60):

   • $f(x) = x - 3$
   • $g(x) = \frac{1}{x}$

3. (61):

   • $f(x) = x^3 + x^2$
   • $g(x) = x^3 - x^2 - x - 4$

4. (65):

   • $f(x) = \frac{1}{x} - x^2$
   • $g(x) = 6x^3 - 5x$

Simplified Steps to Solve:

1. Identify the Dominant Terms: As $x \to \infty$, focus on the terms with the highest power of $x$ because these dominate the behavior of the function.
2. Simplify Expressions: Use the dominant terms to simplify each expression in the limits.
3. Evaluate the Limits: Plug the simplified expressions into the limits to find the asymptotic values.

Would you like me to solve any specific problems from this set, or provide detailed solutions for each? Let me know!

Related Questions:

1. How do you find asymptotic shifts for other types of functions like exponential or logarithmic?
2. What is the importance of identifying dominant terms in asymptotic analysis?
3. How does the concept of limits apply to finding asymptotes in real-world scenarios?
4. What techniques can be used to simplify complex expressions when calculating limits?
5. How do asymptotic behaviors differ for rational versus polynomial functions?

Tip:

Always look for the highest power terms in polynomials when calculating limits as $x \to \infty$. This will make simplifying much easier!