limit of square root of 3+x^2/x as x approaches 0^+

We are tasked with finding the limit of the function \( f(x) = \frac{\sqrt{3 + x^2}}{x} \) as \( x \to 0^+ \).

To solve this:

\[
\lim_{x \to 0^+} \frac{\sqrt{3 + x^2}}{x}
\]

1. At \( x = 0^+ \), the denominator approaches zero, which suggests a potential for an infinite limit. We need to determine the behavior of the function as \( x \) approaches 0 from the right.

2. For small values of \( x \), \( x^2 \) becomes very small, so we approximate \( \sqrt{3 + x^2} \) by factoring out the constant under the square root:

\[
\sqrt{3 + x^2} = \sqrt{3 \left( 1 + \frac{x^2}{3} \right)} \approx \sqrt{3} \cdot \sqrt{1 + \frac{x^2}{3}} \approx \sqrt{3} \left( 1 + \frac{x^2}{6} \right)
\]

This approximation holds for small \( x \), where \( \frac{x^2}{6} \) is negligible. Thus, we can say:

\[
\sqrt{3 + x^2} \approx \sqrt{3}
\]

3. Substituting this approximation into the limit expression:

\[
\lim_{x \to 0^+} \frac{\sqrt{3}}{x}
\]

4. Since \( \frac{\sqrt{3}}{x} \) diverges as \( x \to 0^+ \), the limit is:

\[
\lim_{x \to 0^+} \frac{\sqrt{3}}{x} = +\infty
\]

Thus, the limit of \( \frac{\sqrt{3 + x^2}}{x} \) as \( x \to 0^+ \) is \( +\infty \).

---

Would you like any further clarification or details?

Here are 5 related questions for practice:
1. What is the limit of \( \frac{\sqrt{5 + x^2}}{x} \) as \( x \to 0^+ \)?
2. How would you evaluate \( \lim_{x \to 0^+} \frac{3x + 1}{x} \)?
3. What happens to the function \( \frac{x^2 + 2}{x} \) as \( x \to 0^+ \)?
4. What is the limit of \( \frac{\sqrt{1 + x}}{x} \) as \( x \to 0^+ \)?
5. Can you find \( \lim_{x \to 0^+} \frac{\sqrt{x^2 + 1} - 1}{x} \)?

**Tip:** When dealing with limits involving square roots and small values of \( x \), use approximations like the binomial expansion for efficient estimation.

Explore the limit of the function √(3 + x²) / x as x approaches 0 from the right. Learn about the use of approximation methods and limit laws in calculus to determine that the limit is positive infinity. Suitable for Grades 11-12 or early university students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation