Question
A)) Choose the correct statement. \( \lim _{x \rightarrow 6} \frac{1}{(x-6)^{7}}=-\infty \) \( \lim _{x \rightarrow 6} \frac{1}{(x-6)^{7}}=\infty \) \( \lim _{x \rightarrow 6} \frac{1}{(x-6)^{7}} \) is neither \( -\infty \) nor \( \infty \)
Ask by Dawson Pierce. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The limit \( \lim_{x \rightarrow 6} \frac{1}{(x-6)^7} \) is neither \( -\infty \) nor \( \infty \).
Solution
To determine the correct statement about the limit
\[
\lim_{x \rightarrow 6} \frac{1}{(x-6)^7},
\]
let's analyze the behavior of the function as \( x \) approaches 6 from both sides.
1. **Approaching from the Right (\( x \rightarrow 6^+ \)):**
- When \( x > 6 \), \( (x - 6) \) is a small positive number.
- Raising a small positive number to the 7th power (\( (x-6)^7 \)) remains a small positive number.
- Taking the reciprocal, \( \frac{1}{(x-6)^7} \), results in a very large positive number.
- Therefore, \( \lim_{x \rightarrow 6^+} \frac{1}{(x-6)^7} = +\infty \).
2. **Approaching from the Left (\( x \rightarrow 6^- \)):**
- When \( x < 6 \), \( (x - 6) \) is a small negative number.
- Raising a small negative number to the 7th power (\( (x-6)^7 \)) remains a small negative number (since 7 is odd).
- Taking the reciprocal, \( \frac{1}{(x-6)^7} \), results in a very large negative number.
- Therefore, \( \lim_{x \rightarrow 6^-} \frac{1}{(x-6)^7} = -\infty \).
Since the left-hand limit (\( -\infty \)) and the right-hand limit (\( +\infty \)) are not equal, the two-sided limit does not exist in the traditional sense. Therefore, the correct statement is:
**\( \lim_{x \rightarrow 6} \frac{1}{(x-6)^7} \) is neither \( -\infty \) nor \( \infty \).**
**Answer:**
\( \lim _{x \rightarrow 6} \frac{1}{(x-6)^{7}} \) is neither \( -\infty \) nor \( \infty \)
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Simplify this solution Beyond the Answer
When we evaluate \( \lim_{x \rightarrow 6} \frac{1}{(x-6)^{7}} \), we notice that as \( x \) approaches \( 6 \), \( (x-6)^{7} \) approaches \( 0 \). Since the exponent is odd, when \( x \) approaches \( 6 \) from the left, \( (x-6) \) is negative, resulting in a negative denominator—leading the whole fraction to approach \( -\infty \). Conversely, approaching from the right, \( (x-6) \) is positive, making the fraction approach \( \infty \). So the correct statement here is that the limit diverges based on the direction of approach. You can see how limits can really capture the essence of a function's behavior, and it's fascinating how a single point can lead to such different outcomes!
