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Chapter 2: Problem 5

Find the domain of the function and discuss the behavior of \(f\) near any excluded \(x\) -values. $$f(x)=\frac{1}{x-1}$$

Short Answer

Expert verified
The domain of the function is all real numbers except 1. As \( x \) approaches 1 from the left, \( f(x) \) approaches negative infinity. As \( x \) approaches 1 from the right, \( f(x) \) approaches positive infinity.

Step by step solution

01

Find the domain

The domain of a function is the set of all values of \( x \) which make the function real valued. The denominator of the given function cannot be zero. Hence, solve the equation \( x-1 = 0 \) for \( x \) to find the excluded value. This gives \( x = 1 \). Hence, the domain of the function is all real numbers except 1.
02

Discuss the behavior of \( f \) to the left of excluded \( x \) value

To discuss behavior of \( f \), check the sign of the given function immediately to the left of the excluded \( x \)-value. Substitute a value just less than 1, let's say 0.9 in the function, which gives \( f(0.9) = \frac{1}{0.9-1} = -10 \). Since this is negative, we say that \( f(x) \) approaches negative infinity as \( x \) approaches 1 from the left.
03

Discuss the behavior of \( f \) to the right of excluded \( x \)-value

Similarly, to discuss behavior of \( f \) to the right of 1, substitute a value just more than 1, let's say 1.1 in the function, which gives \( f(1.1) = \frac{1}{1.1-1} = 10 \). Since this is positive, we say that \( f(x) \) approaches positive infinity as \( x \) approaches 1 from the right.

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