Problem 25 - Precalculus Mathematics for Calculus

Question

Find the limit and use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow-1} \frac{x^{2}-x-2}{x^{3}-x}$$

Step-by-Step Solution

Answer

The limit is \(-\frac{3}{2}\).

Step 1: Identify the function

The given limit problem can be rewritten as finding the limit of the function \( f(x) = \frac{x^2 - x - 2}{x^3 - x} \) as \( x \to -1 \).

Step 2: Verify direct substitution

Substitute \( x = -1 \) directly into the function: \( f(-1) = \frac{(-1)^2 -(-1) - 2}{(-1)^3 - (-1)} = \frac{1 + 1 - 2}{-1 + 1} = \frac{0}{0} \). This results in an indeterminate form, so direct substitution cannot be used.

Step 3: Factor and simplify the expression

The numerator \( x^2 - x - 2 \) can be factored as \((x-2)(x+1)\). The denominator \( x^3 - x \) can be factored as \( x(x^2 - 1) \) and further as \( x(x-1)(x+1) \). Thus, the expression becomes: \( f(x) = \frac{(x-2)(x+1)}{x(x-1)(x+1)} \).

Step 4: Cancel common factors

Identify and cancel the common factor \((x+1)\) from the numerator and the denominator: \( f(x) = \frac{(x-2)}{x(x-1)} \) for \( x eq -1 \).

Step 5: Compute the limit of the simplified function

Now, substitute \( x = -1 \) in the simplified function: \( \lim_{x \to -1} \frac{x-2}{x(x-1)} = \frac{-1-2}{-1(-1-1)} = \frac{-3}{2} \).

Step 6: Graphically verify the limit

Using a graphing device, plot \( f(x) = \frac{(x-2)}{x(x-1)} \). As \( x \to -1 \), observe how the function approaches \( \frac{-3}{2} \). This confirms the calculated limit graphically.

Key Concepts

Indeterminate Forms

When tackling limits, especially with expressions that seem straightforward, students often encounter indeterminate forms. An indeterminate form like \( \frac{0}{0} \) doesn't have a clear answer. These forms arise because both the numerator and the denominator evaluate to zero at the point of interest. This doesn't mean the limit doesn’t exist, but rather, a different approach is needed to determine it.

To resolve indeterminate forms, techniques like algebraic manipulation, L'Hôpital's Rule, or factoring can be employed. Factoring allows us to simplify expressions and potentially cancel out terms, which could be contributing to the indeterminate form. Always check for obvious errors in simplification before concluding that a limit cannot be found using standard algebraic techniques.

Factoring Expressions

Factoring is crucial in simplifying expressions to find limits. When faced with an expression like \( \frac{x^2 - x - 2}{x^3 - x} \), factoring helps break the problem into more manageable parts.

For the numerator, \( x^2 - x - 2 \), we search for two numbers that multiply to \(-2\) and add to \(-1\). The solution \((x-2)(x+1)\) factors this expression neatly. Similarly, the denominator \( x^3 - x \) can be split using grouping: first, factor out \(x\), yielding \( x(x^2 - 1) \). Then notice \( x^2 - 1 \) is a difference of squares: \( (x-1)(x+1) \).

Cancelling common factors, such as \( (x+1) \), reduces the complexity of the expression from potentially indeterminate at \( x = -1 \). This simplification is key to finding the actual limit.

Graphical Verification

After algebraically determining the limit, using a graphing tool offers a visual confirmation. Plotting the simplified function \( f(x) = \frac{x-2}{x(x-1)} \) helps us observe its behavior as \( x \to -1 \).

At \( x = -1 \), the original function was indeterminate, but post-simplification, it smoothly approaches a value. The graph should clearly show the function nearing \( \frac{-3}{2} \) as \( x \) approaches \(-1\). If plotted correctly, students will observe a continuous trend towards the calculated limit, reinforcing the algebraic result.

Graphical verifications not only consolidate understanding but also provide a tangible connection between algebraic manipulation and visual interpretation. This dual approach enhances comprehension and confirms correctness of solutions obtained through algebraic methods.