In Exercises 69–82, simplify each complex rational expression. (1/(x+h)² − 1/x²)/h

Verified step by step guidance

1. Step 1: Recognize that the given expression is a complex rational expression. A complex rational expression is a fraction where the numerator and/or denominator contains fractions. The given expression is (1/(x+h)^2 − 1/x^2)/h.
2. Step 2: Simplify the numerator of the complex fraction, which is 1/(x+h)^2 − 1/x^2. To do this, find a common denominator for the two fractions in the numerator. The least common denominator (LCD) of (x+h)^2 and x^2 is (x+h)^2 * x^2.
3. Step 3: Rewrite each fraction in the numerator with the LCD as the denominator. For 1/(x+h)^2, multiply numerator and denominator by x^2 to get x^2/((x+h)^2 * x^2). For 1/x^2, multiply numerator and denominator by (x+h)^2 to get (x+h)^2/((x+h)^2 * x^2).
4. Step 4: Combine the two fractions in the numerator over the common denominator. This gives (x^2 - (x+h)^2)/((x+h)^2 * x^2). Expand (x+h)^2 in the numerator to simplify further.
5. Step 5: Divide the simplified numerator by h, which is the denominator of the original complex fraction. This is equivalent to multiplying the simplified numerator by 1/h. Simplify the resulting expression by factoring and canceling terms where possible.

Key Concepts

Complex Rational Expressions

A complex rational expression is a fraction where the numerator, the denominator, or both contain rational expressions. Simplifying these expressions often involves finding a common denominator, combining terms, and reducing the fraction to its simplest form. Understanding how to manipulate these expressions is crucial for solving problems involving them.

Finding a Common Denominator

To simplify complex rational expressions, it is essential to find a common denominator for the fractions involved. This process allows you to combine the fractions into a single expression, making it easier to simplify. The common denominator is typically the least common multiple (LCM) of the individual denominators.

Limits and Simplification

In calculus and algebra, simplification often involves considering limits, especially when dealing with expressions that approach indeterminate forms. In the given expression, as h approaches zero, understanding how to simplify the expression can help in evaluating limits and understanding the behavior of functions near specific points.