To determine which of the given limit forms are indeterminate, we need to know the conditions for indeterminate forms in calculus. Indeterminate forms arise in limit problems when there's no clear answer without further evaluation. Here’s a breakdown of each option provided:

1. $\frac{0}{k}$, where $k$ is a nonzero constant:

   • This form is not indeterminate. Dividing zero by a nonzero constant results in zero.
   • Not indeterminate.

2. $\infty - \infty$:

   • This form is indeterminate because the difference of two infinities can vary depending on the functions involved, leading to various possible values.
   • Indeterminate.

3. $\frac{0}{0}$:

   • This is an indeterminate form because the limit could approach various values depending on the functions' rates of approaching zero.
   • Indeterminate.

4. $0 \cdot \infty$:

   • This form is indeterminate because the product of zero and infinity could converge to different values depending on the functions involved.
   • Indeterminate.

5. $\infty + \infty$:

   • This form is not indeterminate. Adding two infinities yields infinity.
   • Not indeterminate.

6. $k \cdot \infty$, where $k$ is a nonzero constant:

   • This is not indeterminate. Multiplying a nonzero constant by infinity yields infinity or negative infinity, depending on the sign of $k$.
   • Not indeterminate.

7. $\frac{\infty}{\infty}$:

   • This is an indeterminate form because the limit could approach various values based on the rates of growth of the numerator and denominator.
   • Indeterminate.

Summary

The indeterminate forms in the list are:

• $\infty - \infty$
• $\frac{0}{0}$
• $0 \cdot \infty$
• $\frac{\infty}{\infty}$

Your selection of $\infty - \infty$, $\frac{0}{0}$, and $0 \cdot \infty$ is correct. However, the form $\frac{\infty}{\infty}$ should also be included as an indeterminate form.

Would you like more details on any of these indeterminate forms or have further questions?

Here are five related questions to deepen your understanding:

1. Why is $\frac{0}{0}$ considered indeterminate while $\frac{0}{k}$ is not?
2. How can L'Hôpital's Rule be applied to resolve indeterminate forms?
3. What are examples of limits that result in each type of indeterminate form?
4. How do the functions involved affect the outcome of $\infty - \infty$?
5. Can indeterminate forms lead to finite limits, and if so, how?

Tip: Indeterminate forms often require additional techniques, such as factoring or L'Hôpital's Rule, to evaluate limits accurately.