Question

Evaluate the limit: \(\lim _{x \rightarrow-6} \frac{x^{2}-15 x+54}{x^{2}-6 x}\).

Short answer

The limit is \( \frac{5}{2} \).

Step-by-step solution

Understand the Expression

The expression we need to evaluate is \( \lim _{x \rightarrow -6} \frac{x^{2}-15 x+54}{x^{2}-6 x} \). To evaluate this limit as \( x \) approaches \(-6\), we need to first understand if direct substitution can be used.

Direct Substitution

First, substitute \( x = -6 \) directly into the expression: \( \frac{(-6)^{2} - 15(-6) + 54}{(-6)^{2} - 6(-6)} \). This simplifies to \( \frac{36 + 90 + 54}{36 + 36} \), giving \( \frac{180}{72} \). However, this results in an indeterminate form since both the numerator and denominator evaluate to non-zero values.

Simplify the Expression

Since direct substitution is not feasible, we should simplify the expression first. Factor the numerator and the denominator. The numerator \( x^{2} - 15x + 54 \) can be factored into \((x - 9)(x - 6)\). The denominator \( x^{2} - 6x \) can be factored into \( x(x - 6) \).

Cancel Common Factors

After factoring, the expression becomes \( \frac{(x - 9)(x - 6)}{x(x - 6)} \). Cancel out the common factor \((x - 6)\) from the numerator and the denominator. This simplifies the expression to \( \frac{x - 9}{x} \).

Evaluate the Simplified Limit

Now that we have the simplified expression \( \frac{x - 9}{x} \), substitute \( x = -6 \) into this expression: \( \frac{-6 - 9}{-6} \), which simplifies further to \( \frac{-15}{-6} \). This results in \( \frac{5}{2} \).

Key concepts

Indeterminate Forms

In calculus, evaluating limits is crucial to understanding the behavior of functions as they approach specific points. Sometimes, when substituting values into a function, you may encounter something called an "indeterminate form." These are expressions where the limits cannot be immediately determined, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), and they require further analysis to resolve.

In our example, when we substitute \( x = -6 \) into the expression \( \lim _{x \rightarrow -6} \frac{x^{2}-15 x+54}{x^{2}-6 x} \), we initially believe that direct substitution might work. However, both the numerator and the denominator simplify to real numbers that do not create a zero over zero situation, which suggests we have a solvable limit rather than a classic indeterminate form contradiction. This guides us to further simplify the expression rather than relying on a straightforward evaluation.

Factorization

Factorization is a mathematical process where expressions are rewritten as products of their factors. This technique is particularly useful in limits when trying to cancel common factors from a fraction.

In our original exercise, the expression \( x^{2} - 15x + 54 \) in the numerator was factored as \((x - 9)(x - 6)\), and the expression \( x^{2} - 6x \) in the denominator was rewritten as \( x(x - 6) \).

This factorization reveals hidden factors that may allow for the simplification of the expression, which is a critical step in resolving limits that initially appear indeterminate. By identifying and canceling the common terms present in both the numerator and the denominator, the expression is reduced, cleverly bypassing the indeterminate situation.

Substitution Method

The substitution method in calculus is used to solve limits by directly replacing the variable with the value it approaches. When substitution leads to an indeterminate form, we simplify or manipulate the expression to a form where substitution can directly evaluate the limit.

After simplification, the original expression \( \frac{x^{2}-15 x+54}{x^{2}-6 x} \) was reduced to \( \frac{x - 9}{x} \). Now, we can substitute \( x = -6 \) directly and calculate \( \frac{-6 - 9}{-6} \), which resolves without issue.

This approach shows how substitution can become straightforward once we ensure all unnecessary and problematic parts of the expression are handled and reduced.

Simplification of Expressions

Simplifying expressions is a key approach in calculus, especially when dealing with limits. Simplification involves reducing algebraic expressions to their simplest form by performing operations such as combining like terms, factoring, and canceling common factors.

In the problem, after factorization, the expression \( \frac{(x - 9)(x - 6)}{x(x - 6)} \) simplified to \( \frac{x-9}{x} \). This step removed the \( (x-6) \) term, hence simplifying the handling of the limit.

By reducing the expression, it made the limit solvable and removed any pre-existing indeterminate forms, demonstrating the importance of simplification in making complex calculus problems accessible and solvable for students.