It looks like you've uploaded an image. Let me analyze it to extract the relevant mathematical content and provide a solution. Please give me a moment. PIL import Image

Open the uploaded image to inspect its contents

image_path = '/mnt/data/1733554838641ma3ugbjb.jpg' img = Image.open(image_path) img.show()I've opened the image you uploaded, but I need to extract its content to assist you. I will now attempt to analyze the image and retrieve any text or mathematical expressions present. pytesseract

Use pytesseract to extract text from the image

extracted_text = pytesseract.image_to_string(img) extracted_textThe image contains a limit problem that asks to evaluate a limit using L'Hôpital's Rule:

$\lim_{x \to 0} \frac{3x}{\sin(x) - x}$

To solve this, we can apply L'Hôpital's Rule, which is useful when evaluating limits of indeterminate forms like $\frac{0}{0}$.

Steps to Solve:

1. Check the indeterminate form:
   As $x \to 0$, both the numerator $3x$ and the denominator $\sin(x) - x$ approach 0, so we have the indeterminate form $\frac{0}{0}$, allowing us to apply L'Hôpital's Rule.

2. Differentiate the numerator and the denominator:

   • Numerator: $\frac{d}{dx}[3x] = 3$
   • Denominator: $\frac{d}{dx}[\sin(x) - x] = \cos(x) - 1$

3. Evaluate the limit of the new fraction: $\lim_{x \to 0} \frac{3}{\cos(x) - 1}$ As $x \to 0$, $\cos(0) = 1$, so the denominator $\cos(x) - 1 \to 0$. We still have an indeterminate form $\frac{3}{0}$.

4. Conclusion: Since the limit of $\cos(x) - 1$ as $x \to 0$ approaches 0 from the negative side, the limit of the expression $\frac{3}{\cos(x) - 1}$ approaches $-\infty$.

Thus, the limit is:

$\lim_{x \to 0} \frac{3x}{\sin(x) - x} = -\infty$

Would you like further details on the steps or any clarification?