We are tasked with completing the table for the expression $\frac{x^9 - 1}{x - 1}$ at specific $x$ values, then using that to compute limits as $x \to 1$ from both the left and right sides. Let's start by plugging the given values of $x$ into the expression.

The expression simplifies using algebraic techniques or direct substitution. First, notice that $\frac{x^9 - 1}{x - 1}$ is indeterminate at $x = 1$ (since both the numerator and denominator equal zero), so we need to apply limits.

Calculation for the table

We'll evaluate the function at each of the following $x$ values: $0.9$, $0.99$, $0.999$, $1.1$, $1.01$, and $1.001$.

The general expression is:

$f(x) = \frac{x^9 - 1}{x - 1}$

Let's compute the values for each $x$.

• For $x = 0.9$: $f(0.9) = \frac{0.9^9 - 1}{0.9 - 1}$

• For $x = 0.99$: $f(0.99) = \frac{0.99^9 - 1}{0.99 - 1}$

• For $x = 0.999$: $f(0.999) = \frac{0.999^9 - 1}{0.999 - 1}$

• For $x = 1.1$: $f(1.1) = \frac{1.1^9 - 1}{1.1 - 1}$

• For $x = 1.01$: $f(1.01) = \frac{1.01^9 - 1}{1.01 - 1}$

• For $x = 1.001$: $f(1.001) = \frac{1.001^9 - 1}{1.001 - 1}$

I'll compute these values and then proceed with finding the limits.Here are the computed values for $\frac{x^9 - 1}{x - 1}$ at the given points:

$x$	$\frac{x^9 - 1}{x - 1}$
0.9	6.126
0.99	8.648
0.999	8.964
1.1	13.579
1.01	9.369
1.001	9.036

Limits:

a. $\lim_{x \to 1^-} \frac{x^9 - 1}{x - 1} \approx 9$, since as $x$ approaches 1 from the left (values like 0.9, 0.99, 0.999), the function is approaching values around 9.

b. $\lim_{x \to 1^+} \frac{x^9 - 1}{x - 1} \approx 9$, since as $x$ approaches 1 from the right (values like 1.1, 1.01, 1.001), the function also approaches values around 9.

c. $\lim_{x \to 1} \frac{x^9 - 1}{x - 1} = 9$, as the left-hand and right-hand limits both approach 9.

This implies that the limit exists and equals 9.

Would you like more details on these calculations, or have any specific questions?

5 related questions:

1. What happens if we use L'Hopital's Rule to find this limit analytically?
2. How does the value of the limit change for higher powers, e.g., $x^{10}$?
3. What is the geometric interpretation of this limit in terms of a derivative?
4. Why do the values approach 9 as $x$ approaches 1 from both sides?
5. How does this type of limit compare to simpler polynomial limits?

Tip:

For indeterminate forms like $\frac{0}{0}$, L'Hopital's Rule or polynomial factorizations often help simplify the calculation of limits.