Questions: Compute the values of f(x)=(x-7)/(x-1)^2 in the table to the right and use them to determine lim x→1 f(x).

Solution Steps

To determine $\lim_{x \rightarrow 1} f(x)$ for the function $f(x) = \frac{x-7}{(x-1)^2}$, we can evaluate the function at values of $x$ that are close to 1 from both the left and the right. By computing $f(x)$ for values slightly greater than 1 and slightly less than 1, we can observe the behavior of the function as $x$ approaches 1. This will help us estimate the limit.

We have the function $f(x) = \frac{x-7}{(x-1)^2}$. We evaluate this function for values of $x$ approaching 1 from both sides:

• For $x = 1.1$, $f(1.1) \approx -590$

• For $x = 1.01$, $f(1.01) \approx -59900$

• For $x = 1.001$, $f(1.001) \approx -5999000$

• For $x = 1.0001$, $f(1.0001) \approx -599990000$

• For $x = 0.9$, $f(0.9) \approx -610$

• For $x = 0.99$, $f(0.99) \approx -60100$

• For $x = 0.999$, $f(0.999) \approx -6001000$

• For $x = 0.9999$, $f(0.9999) \approx -600010000$

As $x$ approaches 1 from both the left ($x < 1$) and the right ($x > 1$), the values of $f(x)$ become increasingly negative and large in magnitude. This suggests that the function is approaching negative infinity.

Final Answer