Question
Evaluate \( \lim _{x \rightarrow \infty} \frac{x+17}{3-4 x} \). If the corresponding limit is infinity, then insert INF in the blank. If the corresponding limit is negative infinity, then insert -INF in the blank. If the corresponding limit does not exist, then insert DNE in the blank. Otherwise, write the number as an integer or a reduced fraction.
Ask by Ayala Fuentes. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The limit is \(-\frac{1}{4}\).
Solution
To evaluate the limit:
\[
\lim_{x \rightarrow \infty} \frac{x + 17}{3 - 4x}
\]
Let's analyze the behavior of the function as \( x \) approaches infinity.
1. **Leading Terms Dominance**:
- In the numerator, the leading term is \( x \).
- In the denominator, the leading term is \( -4x \).
2. **Simplifying the Expression**:
\[
\frac{x + 17}{3 - 4x} \approx \frac{x}{-4x} = -\frac{1}{4}
\]
3. **Conclusion**:
As \( x \) becomes very large, the lower-order terms (17 in the numerator and 3 in the denominator) become negligible compared to the leading terms. Therefore, the limit is determined by the ratio of the leading coefficients.
\[
\lim_{x \rightarrow \infty} \frac{x + 17}{3 - 4x} = -\frac{1}{4}
\]
**Answer:** \(-\frac{1}{4}\)
Reviewed and approved by the UpStudy tutoring team
Like
error msg
Explain 
Simplify this solution Beyond the Answer
To evaluate the limit \[ \lim _{x \rightarrow \infty} \frac{x+17}{3-4 x}, \] we can first simplify the expression by dividing both the numerator and the denominator by \( x \): \[ \lim _{x \rightarrow \infty} \frac{\frac{x+17}{x}}{\frac{3-4x}{x}} = \lim _{x \rightarrow \infty} \frac{1 + \frac{17}{x}}{\frac{3}{x} - 4}. \] As \( x \) approaches infinity, \( \frac{17}{x} \) approaches 0 and \( \frac{3}{x} \) also approaches 0. Therefore, the limit simplifies to: \[ \lim _{x \rightarrow \infty} \frac{1 + 0}{0 - 4} = \frac{1}{-4} = -\frac{1}{4}. \] Thus, the answer is: \[ -\frac{1}{4}. \]
