Based on the image provided, I can assist with clarifying and correcting the mathematics.

1. Short Answer (1 mark): The question asks which form is not an indeterminate form.

   • Indeterminate forms include $0/0, \infty/\infty, 0 \cdot \infty, \infty - \infty, 1^\infty, \infty^0, \text{and } 0^0$.
   • In this case, the form $\infty^0$ is not an indeterminate form since it evaluates to a specific limit depending on the base.

2. Long Answer (12 marks):

   • The task involves computing the limit $\lim_{x \to \infty} (1 + x)^{1/x}$.
   • Steps to correct the problem:
     1. Rewrite using logarithms: $y = (1 + x)^{1/x} \implies \ln(y) = \frac{1}{x} \ln(1 + x).$
     2. The limit transforms into: $\lim_{x \to \infty} \ln(y) = \lim_{x \to \infty} \frac{\ln(1 + x)}{x}.$
     3. Use L'Hôpital's Rule:
        • Differentiate the numerator and denominator: $\lim_{x \to \infty} \frac{\ln(1 + x)}{x} = \lim_{x \to \infty} \frac{\frac{1}{1+x}}{1} = \lim_{x \to \infty} \frac{1}{x+1}.$
        • This simplifies to 0.
     4. Conclude: $\ln(y) = 0 \implies y = e^0 = 1.$
     • The correct answer is $\lim_{x \to \infty} (1 + x)^{1/x} = 1$.

Would you like me to elaborate further on any part?

Follow-up Questions

1. Why is $\infty^0$ not considered an indeterminate form in this case?
2. Could $\ln(y)$ approach infinity for a different function setup?
3. How can we systematically use L’Hôpital’s Rule to solve limits?
4. What are the common mistakes made in limit computations like this one?
5. How does logarithmic transformation simplify complex limits?

Tip

Always verify indeterminate forms