https://file.aiquickdraw.com/mathbot/file/17333399296970ivjncd5.jpg

 PIL import Image
import pytesseract

# Load the image to process text
image_path = '/mnt/data/17333399296970ivjncd5.jpg'
extracted_text = pytesseract.image_to_string(Image.open(image_path))

extracted_textThe image text extraction did not yield results, possibly due to low contrast or handwriting style. I'll analyze the mathematical content in the image manually to provide a solution.

The image shows a limit problem involving trigonometric and logarithmic functions. Allow me to break it down into steps and provide a clear explanation.### Solution for the given limit problem

#### Problem Setup:
The problem in the image is structured as:
\[
\lim_{x \to 0} \frac{\left(e^{\frac{x^2}{2}} - 1\right)\sin\left(1 - \cos(x)\right)}{\log\left(1 + x^3 + x^4\right)}
\]

#### Steps to Solve:

1. **Expand the numerator terms using Taylor Series approximations:**
   - For \(e^{\frac{x^2}{2}}\), the Taylor expansion around \(x = 0\) gives:
     \[
     e^{\frac{x^2}{2}} \approx 1 + \frac{x^2}{2}
     \]
     Hence, \(e^{\frac{x^2}{2}} - 1 \approx \frac{x^2}{2}\).

   - For \(1 - \cos(x)\), using the Taylor expansion of \(\cos(x)\) around \(x = 0\), we get:
     \[
     \cos(x) \approx 1 - \frac{x^2}{2}
     \]
     Thus, \(1 - \cos(x) \approx \frac{x^2}{2}\).

     Using the small-angle approximation \(\sin(y) \approx y\) for small \(y\), we have:
     \[
     \sin(1 - \cos(x)) \approx \sin\left(\frac{x^2}{2}\right) \approx \frac{x^2}{2}.
     \]

2. **Expand the denominator term:**
   - For \(\log(1 + x^3 + x^4)\), using the Taylor expansion of \(\log(1 + u) \approx u\) for small \(u\), we get:
     \[
     \log(1 + x^3 + x^4) \approx x^3 + x^4.
     \]

3. **Substitute approximations into the limit:**
   Substituting these expansions into the original limit expression:
   \[
   \lim_{x \to 0} \frac{\left(e^{\frac{x^2}{2}} - 1\right)\sin\left(1 - \cos(x)\right)}{\log\left(1 + x^3 + x^4\right)}
   \approx \lim_{x \to 0} \frac{\left(\frac{x^2}{2}\right)\left(\frac{x^2}{2}\right)}{x^3 + x^4}.
   \]

4. **Simplify the limit:**
   Simplify the numerator and denominator:
   \[
   \frac{\left(\frac{x^2}{2}\right)\left(\frac{x^2}{2}\right)}{x^3 + x^4} = \frac{\frac{x^4}{4}}{x^3 + x^4}.
   \]
   Factor \(x^3\) from the denominator:
   \[
   \frac{\frac{x^4}{4}}{x^3(1 + x)} = \frac{\frac{x}{4}}{1 + x}.
   \]

5. **Evaluate the limit as \(x \to 0\):**
   As \(x \to 0\), the term \(1 + x \to 1\). Therefore:
   \[
   \lim_{x \to 0} \frac{\frac{x}{4}}{1 + x} = \frac{0}{1} = 0.
   \]

### Final Answer:
\[
\boxed{0}
\]

---

Would you like further clarification or detailed explanations for any part of the solution? Here are some related questions for practice:

1. What is the Taylor series expansion for \(e^x\) and \(\sin(x)\)?
2. How does small-angle approximation work for trigonometric functions?
3. How do you apply L’Hôpital’s Rule to solve limits involving indeterminate forms?
4. Can you explain why higher-order terms in the Taylor expansion are ignored for small \(x\)?
5. What are common approximations for \(\log(1 + x)\) and \((1 + x)^n\) for small \(x\)?

**Tip:** Always simplify expressions using Taylor expansions when dealing with small \(x\) in limit problems!

What is the solution to the limit problem \(\lim_{x \to 0} \frac{\left(e^{\frac{x^2}{2}} - 1\right)\sin\left(1 - \cos(x)\right)}{\log\left(1 + x^3 + x^4\right)}\)?

Explore a step-by-step solution to the limit problem \(\lim_{x \to 0} \frac{\left(e^{\frac{x^2}{2}} - 1\right)\sin\left(1 - \cos(x)\right)}{\log\left(1 + x^3 + x^4\right)}\). Learn about Taylor expansions, small-angle approximations, and logarithmic properties. Suitable for calculus students.

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