The problem asks you to evaluate the limit numerically or state that it does not exist:

$\lim_{{x \to 10}} \frac{1}{{(x - 10)^3}}$

Step-by-step solution:

1. The expression inside the limit is $\frac{1}{(x - 10)^3}$.
2. As $x$ approaches 10, the term $(x - 10)$ approaches 0.
3. Since $(x - 10)$ is in the denominator and cubed, the behavior of the function depends on how $(x - 10)$ approaches 0:
   • If $x \to 10^+$ (from the right), $(x - 10)$ will be a small positive number, and $(x - 10)^3$ will be a small positive number. So, $\frac{1}{(x - 10)^3}$ will approach $+\infty$.
   • If $x \to 10^-$ (from the left), $(x - 10)$ will be a small negative number, and $(x - 10)^3$ will be a small negative number. So, $\frac{1}{(x - 10)^3}$ will approach $-\infty$.

Conclusion:

The left and right-hand limits are different ($+\infty$ and $-\infty$), so the limit does not exist (DNE).

Would you like more details on this explanation?

Here are some related questions you can explore:

1. What is the difference between left-hand and right-hand limits?
2. How do we handle limits that approach infinity?
3. What is the behavior of rational functions near vertical asymptotes?
4. How can limits at infinity differ from finite limits?
5. When does the cubic power of a function affect the limit’s behavior?

Tip: Always check for one-sided limits when dealing with expressions that involve division by a term approaching zero.