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.