Questions: Write the function f(x)= (x-1)/(x+1), if x>1; (x+1)/(x-1), if x<1 as a simpler piece-wise defined function and graph it on the set of coordinate axes provided. Use your graph compute each quantity graphically: (a) lim f(x) = (b) lim f(x) = (c) lim f(x) = (d) f(1) = x->-1+ x->1- x->oo

Write the function f(x)= (x-1)/(x+1), if x>1; (x+1)/(x-1), if x<1 as a simpler piece-wise defined function and graph it on the set of coordinate axes provided.

Use your graph compute each quantity graphically:
(a) lim f(x) = (b) lim f(x) = (c) lim f(x) = (d) f(1) =
x->-1+ x->1- x->oo

Transcript text: Write the function f(x)= {(x-1)/(x+1), if x>1; (x+1)/(x-1), if |x|<1} as a simpler piece-wise defined function and graph it on the set of coordinate axes provided. Use your graph compute each quantity graphically: (a) lim f(x) = (b) lim f(x) = (c) lim f(x) = (d) f(1) = x->-1+ x->1- x->oo

Solution

Solution Steps

Step 1: Identify the Piecewise Function

The given function is: \[ f(x) = \begin{cases} \frac{2 - |x|}{x - 1} & \text{if } x \geq 1 \\ \frac{1}{x - 1} & \text{if } x < 1 \end{cases} \]

Step 2: Analyze the Graph for Limits and Function Value

Using the graph provided, we will determine the following:

\(\lim_{x \to 1^-} f(x)\)
\(\lim_{x \to 1^+} f(x)\)
\(\lim_{x \to 1} f(x)\)
\(f(1)\)

Step 3: Determine \(\lim_{x \to 1^-} f(x)\)

From the graph, as \(x\) approaches 1 from the left (\(x < 1\)), the function \(f(x) = \frac{1}{x - 1}\) approaches \(-\infty\).

Step 4: Determine \(\lim_{x \to 1^+} f(x)\)

From the graph, as \(x\) approaches 1 from the right (\(x \geq 1\)), the function \(f(x) = \frac{2 - |x|}{x - 1}\) approaches 1.

Step 5: Determine \(\lim_{x \to 1} f(x)\)

Since the left-hand limit (\(-\infty\)) and the right-hand limit (1) are not equal, the limit \(\lim_{x \to 1} f(x)\) does not exist.

Step 6: Determine \(f(1)\)

From the piecewise function definition, for \(x = 1\), we use the condition \(x \geq 1\): \[ f(1) = \frac{2 - |1|}{1 - 1} = \frac{2 - 1}{0} = \frac{1}{0} \] This is undefined.