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).

Transcript text: Compute the values of $f(x)=\frac{x-7}{(x-1)^{2}}$ in the table to the right and use them to determine $\lim _{x \rightarrow 1} f(x)$.

Solution

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.

Step 1: Evaluate $ f(x) $ for Values Approaching 1

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 $

Step 2: Analyze the Behavior of $ f(x) $

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.