import pytesseract

Load the uploaded image

image_path = '/mnt/data/17331245941275uvg1isq.jpg' text = pytesseract.image_to_string(Image.open(image_path), lang='ind')

I encountered an issue extracting the text from the image. However, based on the visible content, I can analyze and solve the problem regarding vertical asymptotes by using limits.

To determine vertical asymptotes for $f(x) = \frac{x^2 - x - 2}{x+1}$ and $f(x) = \frac{x^2 - x - 2}{x+2}$, we proceed as follows:

a. $f(x) = \frac{x^2 - x - 2}{x+1}$

1. Factorize the numerator: $x^2 - x - 2 = (x - 2)(x + 1)$

2. Simplify the function: $f(x) = \frac{(x - 2)(x + 1)}{x + 1}$

   The $x + 1$ terms cancel out, leaving: f(x) = x - 2 \quad \text{(for x \neq -1)}.

3. Check for vertical asymptotes: The denominator $x + 1 = 0$ when $x = -1$, but since $x + 1$ cancels out, there is no vertical asymptote for this function.

b. $f(x) = \frac{x^2 - x - 2}{x+2}$

1. Factorize the numerator: $x^2 - x - 2 = (x - 2)(x + 1)$

2. Function remains: $f(x) = \frac{(x - 2)(x + 1)}{x + 2}$

3. Check for vertical asymptotes: The denominator $x + 2 = 0$ when $x = -2$, so there is a vertical asymptote at: $x = -2$

Conclusion

• For $f(x) = \frac{x^2 - x - 2}{x+1}$: No vertical asymptote.
• For $f(x) = \frac{x^2 - x - 2}{x+2}$: Vertical asymptote at $x = -2$.

Would you like a step-by-step graph or additional clarification?

Follow-up Questions

1. How do cancellations in rational functions affect vertical asymptotes?
2. What happens when the numerator and denominator have common factors?
3. Can a rational function have more than one vertical asymptote?
4. How do horizontal asymptotes differ from vertical ones?
5. Can we determine slant asymptotes using limits?

Tip: Always simplify the function before analyzing for asymptotes to avoid misinterpretations.