Question

Evaluate the given limit. $$ \lim _{x \rightarrow 0} \frac{x^{2}+2 x}{x^{2}-2 x} $$

Short answer

The limit is -1.

Step-by-step solution

Check the expression at x=0

First, substitute \( x = 0 \) into the expression to see if the function is undefined at this point. This helps determine if simplification or other methods are needed. If we substitute \( x = 0 \) into the expression, we get \( \frac{0^2 + 2 \times 0}{0^2 - 2 \times 0} = \frac{0}{0} \), which means the function is indeterminate at \( x = 0 \). We need further analysis.

Factor the numerator and the denominator

Next, factor both the numerator and the denominator of the expression. The numerator \( x^2 + 2x \) factors to \( x(x + 2) \). The denominator \( x^2 - 2x \) factors to \( x(x - 2) \). Thus, the expression becomes: \[ \frac{x(x + 2)}{x(x - 2)} \]

Simplify the expression

Since the expression \( \frac{x(x + 2)}{x(x - 2)} \) has a common factor \( x \) in both the numerator and the denominator, we can cancel it out. This simplifies the expression to: \[ \frac{x + 2}{x - 2} \]

Evaluate the limit of the simplified expression

Now, we evaluate the limit of the simplified expression as \( x \rightarrow 0 \). Substitute 0 into the expression \( \frac{x+2}{x-2} \), which gives: \[ \frac{0 + 2}{0 - 2} = \frac{2}{-2} = -1 \] Therefore, the limit is \( -1 \).

Key concepts

Indeterminate Forms

When evaluating limits in calculus, we sometimes encounter expressions that seem to have no clear value at a given point. These are called indeterminate forms. An indeterminate form often appears in the context of limits, such as \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), or \( \infty - \infty \). In our exercise, substituting \( x = 0 \) into the fractional expression results in \( \frac{0}{0} \). This situation indicates that direct substitution does not provide a valid result, and further methods, like simplification, are necessary to resolve the uncertainty. Recognizing indeterminate forms is crucial for choosing the right technique to evaluate a limit successfully.

Factorization

Factorization is a mathematical technique used to rewrite expressions as a product of their simpler components. This process is especially useful when dealing with polynomial expressions in calculus. In the given problem, factorization helps address the indeterminate form. By expressing both the numerator \( x^2 + 2x \) and the denominator \( x^2 - 2x \) in factored form, we can identify and cancel out common terms.

• The numerator, \( x^2 + 2x \), factors to \( x(x + 2) \).
• The denominator, \( x^2 - 2x \), factors to \( x(x - 2) \).

Here, factorization reveals the common factor \( x \), simplifying the expression and paving the way for easy limit evaluation.

Rational Functions

Rational functions are quotients of polynomial expressions, and they feature heavily in calculus due to their distinct properties and behaviors. In our exercise, the original expression is a rational function, where both the numerator and denominator are polynomial expressions. Understanding how to manipulate rational functions, such as simplifying them through factoring, is crucial. This simplification aids in evaluating the limits of these functions without ambiguity. Once simplified, these functions tend to present fewer complications when we try to understand their behavior near specific points, such as the limit point in this problem.

Simplification in Calculus

Simplification is an essential tool in calculus, especially when evaluating limits. By reducing complex expressions to simpler forms, we make calculations more straightforward and manageable. In this exercise, simplification happens after factoring the expression. We cancel out the common factor \( x \) from both the numerator and denominator, reducing the original expression \( \frac{x(x + 2)}{x(x - 2)} \) to \( \frac{x + 2}{x - 2} \). This simplification enables us to directly substitute \( x = 0 \) and find the limit without facing indeterminacy. Simplification not only helps solve limits but also improves understanding of underlying mathematical properties.