We want to evaluate the limit:

\[
\lim_{x \to \infty} \frac{\sqrt{x^2 - 7}}{x - 3}
\]

### Step 1: Simplify the expression inside the limit
Notice that as \( x \) approaches infinity, \( x^2 \) dominates over 7, so \( \sqrt{x^2 - 7} \) can be approximated by \( \sqrt{x^2} = |x| \). Since \( x \) is positive as \( x \) approaches infinity, \( |x| = x \). Therefore, we approximate:

\[
\sqrt{x^2 - 7} \approx x
\]

### Step 2: Simplify the limit expression
Substituting the approximation into the limit:

\[
\lim_{x \to \infty} \frac{\sqrt{x^2 - 7}}{x - 3} \approx \lim_{x \to \infty} \frac{x}{x - 3}
\]

### Step 3: Divide numerator and denominator by \( x \)
Simplify the expression by dividing both the numerator and the denominator by \( x \):

\[
\lim_{x \to \infty} \frac{x}{x(1 - \frac{3}{x})} = \lim_{x \to \infty} \frac{1}{1 - \frac{3}{x}}
\]

### Step 4: Evaluate the limit
As \( x \) approaches infinity, \( \frac{3}{x} \) approaches 0:

\[
\lim_{x \to \infty} \frac{1}{1 - 0} = \frac{1}{1} = 1
\]

Thus, the limit is:

\[
\boxed{1}
\]

Would you like further details or have any other questions?

Here are some related questions you might consider:
1. What happens if the limit is taken as \( x \) approaches negative infinity?
2. How would the limit change if we replaced \( -7 \) with another constant?
3. Can you generalize this approach for any expression of the form \( \sqrt{x^2 + C} \) over \( x + D \)?
4. What happens if you consider the limit where the denominator is \( x + 3 \) instead of \( x - 3 \)?
5. How do you evaluate the limit if the square root term had a higher power of \( x \) (e.g., \( \sqrt{x^4 - 7} \))?

**Tip:** When dealing with limits that involve square roots of polynomials, it's often helpful to factor out the highest power of \( x \) within the square root to simplify the expression.

Learn how to evaluate the limit of sqrt(x^2 - 7) / (x - 3) as x approaches infinity with a detailed step-by-step solution. This problem covers key concepts in limits, infinity, and square roots. Suitable for advanced high school students and anyone interested in calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation