The problem you provided asks to find the main form of $Cx^\alpha$ as $x \to 0$ for the following functions:

• (a) $f(x) = \sqrt{1 - 2x - 1 + x}$
• (b) $f(x) = \tan(x) - \sin(x)$
• (c) $f(x) = e^{x^2} - \cos(x)$
• (d) $f(x) = \sqrt{3} - \sqrt{2 + \cos(x)}$

This involves expanding each function using series expansions for small values of $x$ (i.e., $x \to 0$) and identifying the dominant behavior. Let's go through each one to analyze the asymptotic form.

I will start by simplifying the functions and expanding them. Let me proceed with the calculations now. sympy as sp

Define the variable

x = sp.symbols('x')

Define the functions

f_a = sp.sqrt(1 - 2*x - 1 + x) # Function (a) f_b = sp.tan(x) - sp.sin(x) # Function (b) f_c = sp.exp(x**2) - sp.cos(x) # Function (c) f_d = sp.sqrt(3) - sp.sqrt(2 + sp.cos(x)) # Function (d)

Perform series expansion at x = 0 for each function

exp_a = sp.series(f_a, x, 0, 5).simplify() exp_b = sp.series(f_b, x, 0, 5).simplify() exp_c = sp.series(f_c, x, 0, 5).simplify() exp_d = sp.series(f_d, x, 0, 5).simplify()

exp_a, exp_b, exp_c, exp_dHere are the results of the series expansions for each function as $x \to 0$:

1. For $f(x) = \sqrt{1 - 2x - 1 + x}$: $f(x) = i\sqrt{x}$ This function involves an imaginary term, suggesting it has a complex nature for small $x$.

2. For $f(x) = \tan(x) - \sin(x)$: $f(x) = \frac{x^3}{2} + O(x^5)$ The dominant term is $\frac{x^3}{2}$, so the function behaves like $Cx^\alpha$ with $\alpha = 3$.

3. For $f(x) = e^{x^2} - \cos(x)$: $f(x) = \frac{3x^2}{2} + \frac{11x^4}{24} + O(x^5)$ The dominant term is $\frac{3x^2}{2}$, so the function behaves like $Cx^\alpha$ with $\alpha = 2$.

4. For $f(x) = \sqrt{3} - \sqrt{2 + \cos(x)}$: $f(x) = \frac{\sqrt{3}x^2}{12} - \frac{\sqrt{3}x^4}{288} + O(x^5)$ The dominant term is $\frac{\sqrt{3}x^2}{12}$, so the function behaves like $Cx^\alpha$ with $\alpha = 2$.

Summary of results:

• (a) $f(x) = \sqrt{1 - 2x - 1 + x}$ tends to an imaginary term $i\sqrt{x}$.
• (b) $f(x) = \tan(x) - \sin(x)$ behaves like $x^3$.
• (c) $f(x) = e^{x^2} - \cos(x)$ behaves like $x^2$.
• (d) $f(x) = \sqrt{3} - \sqrt{2 + \cos(x)}$ behaves like $x^2$.

Would you like further clarification on any part of the solution?

Related questions to explore:

1. How does the behavior of $\tan(x) - \sin(x)$ change as $x \to 0$?
2. What is the significance of the imaginary term in function (a)?
3. Can we generalize the form $Cx^\alpha$ for more complicated functions?
4. How do higher-order terms in the series affect the asymptotic behavior of these functions?
5. What is the connection between Taylor series and asymptotic analysis?

Tip: The series expansion helps identify the most significant terms for small values of $x$, which are critical for understanding the behavior near zero.