The task in the image asks to find the leading term of each function as $x \to +\infty$ and determine the growth order of these functions. Let's go through each one step by step.

Given functions:

1. $a) \ x^2 + 100x + 10000$
2. $b) \ \frac{2x^5}{x^3 - 3x + 1}$
3. $c) \ \sqrt[3]{x^2 - x + \sqrt{x}}$
4. $d) \ \sqrt{1 + \sqrt{1 + \sqrt{x}}}$

Analyzing each function as $x \to +\infty$:

1. $a) \ x^2 + 100x + 10000$

For large $x$, the highest degree term dominates. In this case, $x^2$ grows faster than $100x$ and $10000$.

• Leading term: $x^2$
• Growth order: $x^2$

2. $b) \ \frac{2x^5}{x^3 - 3x + 1}$

For large $x$, the dominant term in the numerator is $2x^5$, and the dominant term in the denominator is $x^3$.

• Simplifying for large $x$: $\frac{2x^5}{x^3 - 3x + 1} \approx \frac{2x^5}{x^3} = 2x^2$
• Leading term: $2x^2$
• Growth order: $x^2$

3. $c) \ \sqrt[3]{x^2 - x + \sqrt{x}}$

For large $x$, the highest degree term inside the cube root is $x^2$.

• Approximating for large $x$: [ \sqrt[3]{x^2 - x + \sqrt{x}} \approx \sqrt[3]{x^2} ] $\sqrt[3]{x^2} = x^{2/3}$
• Leading term: $x^{2/3}$
• Growth order: $x^{2/3}$

4. $d) \ \sqrt{1 + \sqrt{1 + \sqrt{x}}}$

For large $x$, the dominant term inside the square roots will be the innermost square root $\sqrt{x}$.

• Approximating for large $x$: $\sqrt{1 + \sqrt{1 + \sqrt{x}}} \approx \sqrt{1 + \sqrt{\sqrt{x}}} \approx \sqrt{\sqrt{x}} = x^{1/4}$
• Leading term: $x^{1/4}$
• Growth order: $x^{1/4}$

Summary of Leading Terms and Growth Orders:

• (a): Leading term: $x^2$, Growth order: $x^2$
• (b): Leading term: $2x^2$, Growth order: $x^2$
• (c): Leading term: $x^{2/3}$, Growth order: $x^{2/3}$
• (d): Leading term: $x^{1/4}$, Growth order: $x^{1/4}$

Let me know if you'd like more details on any of these steps or have any questions!

Relative questions:

1. How does the growth order of a polynomial compare to an exponential function?
2. What happens if we compare functions with logarithmic growth versus polynomial growth?
3. How can L'Hopital's rule help in analyzing limits involving infinity?
4. What are the properties of cube roots and square roots in limit calculations?
5. How would the growth change if we had additional terms involving $\ln(x)$ in these functions?

Tip:

When analyzing functions as $x \to \infty$, always focus on the highest-degree terms because they dominate the behavior of the function in this limit.