Question

, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 2^{-}} \frac{x^{2}-x-2}{|x-2|} $$

Short answer

The limit is \(-3\).

Step-by-step solution

Understand the limit approach

The given limit is a one-sided limit where we need to find the value of the expression as \( x \) approaches 2 from the left (indicated by \( x \to 2^- \)). Here, \( |x-2| \) always evaluates to a positive value close to 0 for \( x < 2 \).

Simplify the numerator

The numerator of the fraction is \( x^2 - x - 2 \). To simplify, factor it: \( x^2 - x - 2 = (x - 2)(x + 1) \).

Rewrite and simplify the expression

The entire expression becomes \( \frac{(x-2)(x+1)}{|x-2|} \). For \( x < 2 \), \( |x-2| = -(x-2) \) since it is negative. Thus, the expression can be rewritten as \( \frac{(x-2)(x+1)}{-(x-2)} \).

Simplify further by canceling out terms

In the expression \( \frac{(x-2)(x+1)}{-(x-2)} \), the \( (x-2) \) terms cancel each other, resulting in \( -(x+1) \).

Evaluate the limit as x approaches 2 from the left

Now, substitute \( x = 2 \) into the simplified expression \(-(x+1)\), giving \(-(2+1) = -3\). Thus, the limit is \(-3\).

Key concepts

One-sided Limits

One-sided limits are a fundamental concept in calculus. They help us to understand how a function behaves as it approaches a particular point from one side only. Specifically, in one-sided limits, there are two variants: left-hand limits (denoted as \( x \to a^{-} \) as a point is approached from the left) and right-hand limits (denoted as \( x \to a^{+} \) as a point is approached from the right).
When working with one-sided limits:

• It is important to note which side you are approaching from, as it affects how you interpret the values involved.
• You focus solely on the direction specified (left or right) to determine how the function behaves.
• In many problems, especially those involving absolute values, one-sided limits help resolve indeterminacies.

Understanding and calculating one-sided limits is essential for accurately analyzing functions at points where their behavior might change abruptly.

Limit Evaluation Techniques

Finding limits often involves a variety of techniques, depending on the form and complexity of the function. Here are some common methods used in limit evaluation:

• Direct Substitution: The simplest approach where you substitute the point directly into the function, used mostly when no indeterminate forms such as 0/0 arise.
• Factoring: Often used when the function is a polynomial; factoring can cancel out terms leading to determinable values.
• Rationalization: Helpful when dealing with irrational expressions in the numerator or denominator.
• Limits involving infinity: Sometimes require understanding asymptotic behavior such as horizontal or vertical asymptotes.
• Using the absolute value: For functions that include absolute values, understanding their property to split the function into distinct cases helps.

Mastering several of these techniques provides a toolkit for approaching a wide range of limit problems that may arise in calculus.

Absolute Value in Limits

The absolute value function is pivotal in interpreting a limit, especially for non-differentiable points. Absolute value, by definition, provides positive values regardless of input, transforming expressions differently based on the input's sign. Here's how it affects limits:

• Piecewise Definition: The absolute value can be defined as a piecewise function as \( |x| = x \) when \( x \geq 0 \) and \( |x| = -x \) when \( x < 0 \).
• Influencing Limits: In the context of limits, it splits the approach into cases such as left-hand and right-hand. This division allows evaluating the limit by handling each case selectively.
• Handling Discontinuities: It aids in handling discontinuities and changes in sign, allowing smoother passage through these transitions in limit evaluations.

Effectively understanding how to manipulate absolute values is crucial when handling functions that behave differently across specific boundaries.

Factoring in Algebra

Factoring is a powerful tool in algebra that simplifies expressions and resolves indeterminate forms in limits. It involves rewriting an expression as a product of its factors, which often simplifies complicated expressions. Here's how factoring plays into limits:

• Simplification: Factoring can break down a complex polynomial into simpler, manageable parts, which can be canceled out if common, leading to a reduction of terms.
• Resolution of Indeterminate Forms: Often, expressions like \( \frac{0}{0} \) in limits transform into determinable values through factoring since these forms usually mean a common factor can cancel.
• Identifying Critical Points: Factoring reveals critical points and roots of the polynomial essential in evaluating functions at specific points.

In calculus, being proficient in algebraic techniques such as factoring is key to simplifying expressions and solving many types of problems efficiently.