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Chapter 2: Problem 7

Find the domain of the function and discuss the behavior of \(f\) near any excluded \(x\) -values. $$f(x)=\frac{3 x^{2}}{x^{2}-1}$$

Short Answer

Expert verified
The domain of the function \(f(x)=\frac{3 x^{2}}{x^{2}-1}\) is all real numbers except -1 and 1. The function tends to \(-鈭瀄) as \(x\) approaches 1 from the left and \(x\) approaches -1 from the right, and tends to \(+鈭瀄) as \(x\) approaches 1 from the right and \(x\) approaches -1 from the left.

Step by step solution

01

Finding the Denominator's Zeros

Set the denominator equal to zero and solve for \(x\):\n\(x^2 - 1 = 0\)\nThis can be factorized as \((x+1)(x-1) = 0\).\nThus, the solutions are \(x = -1\) and \(x = 1\). These are the values for which \(f(x)\) is undefined.
02

Determining the Domain

Except for these two points, the function \(f(x)\) is defined for all real numbers. Thus, the domain of \(f(x)\) is \(x 鈭� \mathbb{R}\) except \(x = -1\) and \(x = 1\).
03

Discussing the Limiting Behavior

To understand the behavior of the function as \(x\) approaches these excluded values, we examine the limit of \(f(x)\) as \(x\) approaches these points.\n\nAs \(x\) approaches 1 from the left, the denominator of \(f(x)\) approaches 0 from the negative side, this implies that the function goes to negative infinity.\n\nAs \(x\) approaches 1 from the right, the denominator of \(f(x)\) approaches 0 from the positive side, implying that the function goes to positive infinity.\n\nA similar analysis can be performed for \(x = -1\), resulting in the same conclusion. The function takes on large negative values as \(x\) approaches -1 from the left and large positive values as \(x\) approaches -1 from the right.

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