Evaluate each of the following limits, then identify any vertical asymptotes of the function *f*(*x*)=1(*x*−10)6 .

lim*x*→10−1(*x*−10)6lim*x*→10+1(*x*−10)6lim*x*→101(*x*−10)6



We are tasked with evaluating the following limits for the function \( f(x) = \frac{1}{(x - 10)^6} \):

1. **Left-hand limit**: 
   \[
   \lim_{x \to 10^-} \frac{1}{(x - 10)^6}
   \]
   - As \( x \to 10^- \), \( x - 10 \) approaches a small negative number. Since this expression is raised to the sixth power (an even number), \( (x - 10)^6 \) will be positive and very small. The reciprocal of a very small positive number will approach infinity.
   - Therefore,
   \[
   \lim_{x \to 10^-} \frac{1}{(x - 10)^6} = +\infty.
   \]

2. **Right-hand limit**:
   \[
   \lim_{x \to 10^+} \frac{1}{(x - 10)^6}
   \]
   - As \( x \to 10^+ \), \( x - 10 \) approaches a small positive number. Again, since this is raised to the sixth power, \( (x - 10)^6 \) will be positive and very small. The reciprocal of a very small positive number will approach infinity.
   - Therefore,
   \[
   \lim_{x \to 10^+} \frac{1}{(x - 10)^6} = +\infty.
   \]

3. **Two-sided limit**:
   \[
   \lim_{x \to 10} \frac{1}{(x - 10)^6}
   \]
   - Since both the left-hand and right-hand limits approach \( +\infty \), the two-sided limit will also approach infinity.
   - Therefore,
   \[
   \lim_{x \to 10} \frac{1}{(x - 10)^6} = +\infty.
   \]

### Vertical Asymptote:
A vertical asymptote occurs when the function approaches \( \infty \) (or \( -\infty \)) as \( x \) approaches a certain value. From the above calculations, as \( x \to 10 \), the function approaches \( +\infty \) from both sides. This means there is a vertical asymptote at \( x = 10 \).

### Summary:
- \( \lim_{x \to 10^-} \frac{1}{(x - 10)^6} = +\infty \)
- \( \lim_{x \to 10^+} \frac{1}{(x - 10)^6} = +\infty \)
- \( \lim_{x \to 10} \frac{1}{(x - 10)^6} = +\infty \)

There is a vertical asymptote at \( x = 10 \).

Would you like further clarification or additional questions? Here are 5 related questions:
1. What is the behavior of the function \( f(x) = \frac{1}{(x - 10)^2} \) as \( x \to 10 \)?
2. How can we identify vertical asymptotes in rational functions?
3. What happens to the limits of a function when the denominator approaches zero?
4. How would you graph a function with a vertical asymptote?
5. What are some common types of discontinuities in functions?

**Tip**: To determine vertical asymptotes, look for values of \( x \) that make the denominator of a rational function equal to zero.

Evaluate each of the following limits, then identify any vertical asymptotes of the function f(x) = 1/(x-10)^6: lim x→10− 1/(x−10)^6, lim x→10+ 1/(x−10)^6, lim x→10 1/(x−10)^6

Learn how to evaluate the limits and identify vertical asymptotes for the function f(x) = 1/(x-10)^6. This problem demonstrates how limits behave as x approaches a critical value, resulting in a vertical asymptote at x = 10. Suitable for advanced high school students (Grades 11-12).

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