https://file.aiquickdraw.com/mathbot/file/1725519935640blwejq58.jpg

The limit in the image asks us to find the limit of a function \( y \) as \( x \to \infty \). The given function is:

\[
y = \frac{\left( 12 - \frac{5}{x} \right) - \frac{14}{\sqrt{x}}}{3 + \frac{7}{x}} \quad \text{as} \quad x \to \infty
\]

### Step-by-step solution:

1. **Simplifying the expression**: 
   When \( x \to \infty \), the terms \( \frac{5}{x} \), \( \frac{14}{\sqrt{x}} \), and \( \frac{7}{x} \) tend to 0 because any constant divided by infinity approaches 0.

2. **Evaluating each part at \( x \to \infty \)**:
   \[
   \lim_{x \to \infty} \left( 12 - \frac{5}{x} \right) = 12
   \]
   \[
   \lim_{x \to \infty} \frac{14}{\sqrt{x}} = 0
   \]
   \[
   \lim_{x \to \infty} \left( 3 + \frac{7}{x} \right) = 3
   \]

3. **Substituting these into the original expression**:
   \[
   y = \frac{12 - 0}{3 + 0} = \frac{12}{3} = 4
   \]

### Conclusion:
\[
\lim_{x \to \infty} y = 4
\]

Would you like further details or have any questions on this solution?

Here are some related questions you might find useful:
1. How do limits behave when there are square roots involved?
2. What happens to terms like \( \frac{1}{x} \) as \( x \to \infty \)?
3. How can we simplify complex rational functions when finding limits?
4. How do limits apply in real-world scenarios like growth rates?
5. Can limits at infinity lead to indeterminate forms, and how do we resolve them?

**Tip**: When dealing with limits as \( x \to \infty \), always check which terms vanish by analyzing their denominators. The larger the denominator, the quicker the term vanishes as \( x \to \infty \).

Find the limit of the function y = (12 - 5/x - 14/√x) / (3 + 7/x) as x → ∞.

Learn how to find the limit of a rational function involving terms like 12 - 5/x and 14/√x as x approaches infinity. The problem uses limit laws and simplifications to evaluate the function's behavior at large values of x.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation