Let's analyze the limits provided in Question 11 to determine which one is of indeterminate form. The question asks which limit results in an indeterminate form, commonly $\frac{0}{0}$, $\frac{\infty}{\infty}$, or similar.

Here are the options with analysis:

1. $\lim_{x \to 0} e^x \sin(x)$

   • As $x \to 0$, $e^x \to 1$ and $\sin(x) \to 0$.
   • Therefore, $e^x \sin(x) \to 1 \cdot 0 = 0$.
   • This limit is not indeterminate, it evaluates to 0.

2. $\lim_{x \to 0} \frac{\cos(x)}{x}$

   • As $x \to 0$, $\cos(x) \to 1$, and the denominator $x \to 0$.
   • This forms a $\frac{1}{0}$ type, which is not indeterminate but tends to infinity.
   • So, this is not an indeterminate form.

3. $\lim_{x \to 0^+} \sin(x) \ln(x)$

   • As $x \to 0^+$, $\sin(x) \to 0$ and $\ln(x) \to -\infty$.
   • This forms $0 \cdot (-\infty)$, which is indeterminate because the product of 0 and infinity is not immediately clear.
   • This is an indeterminate form.

4. $\lim_{x \to 0} \frac{\cos(x) - 1}{e^x}$

   • As $x \to 0$, $\cos(x) - 1 \to 0$ and $e^x \to 1$.
   • This forms $\frac{0}{1}$, which equals 0.
   • Therefore, this is not indeterminate.

Conclusion:

The third option $\lim_{x \to 0^+} \sin(x) \ln(x)$ is the indeterminate form.

Would you like more details or have any questions? Here are some related topics you might explore:

1. What are the common types of indeterminate forms?
2. How does L'Hopital's Rule help with indeterminate forms?
3. What are the steps to solve limits that involve logarithmic functions?
4. How do trigonometric limits behave as $x \to 0$?
5. What are some methods to evaluate difficult limits?

Tip: Indeterminate forms like $0 \cdot (-\infty)$ can often be transformed using algebraic manipulation or by taking logarithms to simplify the expression for better evaluation.