Let f (x) =
√x2 + 1
x2 − 3x + 2 . Find all vertical asymptotes of f and justify your answer by taking the
appropriate limit(s)

We are given the function \( f(x) = \frac{\sqrt{x^2 + 1}}{x^2 - 3x + 2} \), and we need to find all vertical asymptotes of this function by taking the appropriate limits.

### Step 1: Identify where the denominator is zero
A vertical asymptote occurs when the denominator of a rational function is zero and the numerator is nonzero at that point.

The denominator is:
\[
x^2 - 3x + 2 = 0
\]

We can factor this quadratic equation:
\[
x^2 - 3x + 2 = (x - 1)(x - 2)
\]

Thus, the denominator equals zero when \( x = 1 \) or \( x = 2 \). These are potential vertical asymptotes.

### Step 2: Check the behavior of \( f(x) \) as \( x \to 1 \) and \( x \to 2 \)

We now examine the limits as \( x \to 1 \) and \( x \to 2 \) to confirm whether these points lead to vertical asymptotes.

#### As \( x \to 1 \):
\[
f(x) = \frac{\sqrt{x^2 + 1}}{(x - 1)(x - 2)}
\]
As \( x \to 1 \), the denominator approaches \( 0 \) because of the factor \( (x - 1) \), while the numerator remains finite (since \( \sqrt{x^2 + 1} \to \sqrt{1^2 + 1} = \sqrt{2} \)). Therefore, the limit as \( x \to 1 \) is of the form:
\[
\lim_{x \to 1} f(x) = \pm \infty
\]
This confirms that there is a vertical asymptote at \( x = 1 \).

#### As \( x \to 2 \):
Similarly, as \( x \to 2 \), the denominator approaches zero because of the factor \( (x - 2) \), while the numerator remains finite:
\[
\lim_{x \to 2} f(x) = \pm \infty
\]
This confirms that there is also a vertical asymptote at \( x = 2 \).

### Conclusion:
The function \( f(x) = \frac{\sqrt{x^2 + 1}}{x^2 - 3x + 2} \) has vertical asymptotes at \( x = 1 \) and \( x = 2 \).

Would you like me to clarify any steps or provide additional details?

---

Here are 5 related questions:
1. What is the horizontal asymptote of the function \( f(x) \)?
2. How do we find vertical asymptotes for functions involving square roots?
3. Can a function have a removable discontinuity instead of an asymptote? How would you check for that?
4. What are the limits of the function as \( x \to -\infty \) and \( x \to +\infty \)?
5. How does the factorization of polynomials help in determining vertical asymptotes?

**Tip**: Always check both the numerator and denominator when finding asymptotes; the behavior of both can influence the type of discontinuity at a point.

Let f(x) = √(x^2 + 1) / (x^2 − 3x + 2). Find all vertical asymptotes of f and justify your answer by taking the appropriate limit(s).

Learn how to find the vertical asymptotes of the function f(x) = √(x^2 + 1) / (x^2 − 3x + 2). This solution explores rational functions, limits, and the process of factoring quadratic expressions to locate vertical asymptotes. Suitable for advanced high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation