from sympy import symbols, limit, cos, asin, sqrt, series

Define the variable

x = symbols('x')

Define the expressions

numerator = 5 * x denominator = asin(x)5 inner_expr = 6 / (3 + x2) - sqrt(cos(2 * x)) expr = inner_expr * (numerator / denominator)

Expand using series for Taylor approximations

expanded_inner = series(6 / (3 + x**2) - sqrt(cos(2 * x)), x, 0, 5).removeO() expanded_arcsin = series(asin(x), x, 0, 5).removeO()**5

Simplify the limit calculation

simplified_expr = expanded_inner * (numerator / expanded_arcsin)

Compute the limit

result = limit(simplified_expr, x, 0) resultoooosuccessThe limit of the given expression diverges to infinity ($\infty$). This indicates that the expression becomes unbounded as $x \to 0$.

Let me know if you want a detailed explanation of the steps, or feel free to ask additional questions. Here are 5 related questions for practice:

1. How do Taylor or Maclaurin series help in simplifying limits involving transcendental functions?
2. Can we analyze the individual contributions to the divergence in this expression?
3. What happens if we replace $\arcsin(x)$ with a simpler function, like $x$?
4. How would the series expansion change if a higher-order approximation were used?
5. Can L’Hôpital’s rule provide an alternative approach to verify the divergence?

Tip: Always check if your function has terms with a faster rate of growth (or decay) to detect divergence/convergence tendencies early.