Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \lim _{x \rightarrow 0}\left(\frac{2 x^{2}-x}{x}\right)=-1 $$

Step-by-Step Solution

Answer

The limit is -1.

Step 1: Understand the Problem

We need to find the limit \[ \lim_{x \to 0} \left( \frac{2x^2 - x}{x} \right) \]Evaluate whether the expression simplifies to determine how it behaves as \(x\) approaches 0.

Step 2: Simplify the Expression

Start by factoring out \(x\) from the numerator:\[ 2x^2 - x = x(2x - 1) \]Now rewrite the original limit expression:\[ \lim_{x \to 0} \left( \frac{x(2x - 1)}{x} \right) \]Cancel the \(x\) in the numerator and the denominator:\[ \lim_{x \to 0} (2x - 1) \]

Step 3: Evaluate the Limit

With the simplified expression \(2x - 1\), substitute \(x = 0\):\[ 2(0) - 1 = -1 \]Therefore, the limit of the original expression as \(x\) approaches 0 is -1.

Key Concepts

Limit Evaluation

The concept of 'Limit Evaluation' is fundamental in calculus. A limit describes the value that a function approaches as the input approaches some value. This is pivotal in understanding behaviors of mathematical functions, especially where they might not explicitly meet the typical rules of algebra due to undefined or indeterminate forms.

When evaluating limits, the goal is to find where a function heads as the input becomes infinitesimally close to a given value. We often use this method to understand tendencies and to expose hidden properties of functions. In our exercise, this is achieved by simplifying the expression into a form that provides a direct substitution of the approaching value, in this case, as \( x \) approaches 0.

It's important to note that not all limits require complex manipulation. Sometimes, a simple substitution post-simplification can reveal the limit, as it did when \( 2x - 1 \) simplified to -1 when \( x = 0 \). However, it's crucial to manage the expressions correctly to avoid misleading conclusions.

Simplifying Expressions

Simplifying expressions is an indispensable part of calculus and algebra. The goal here is to make a complex expression easier to handle or compute, which is often achieved by breaking it into simpler or more manageable parts.

In our exercise, the expression \( \frac{2x^2 - x}{x} \) seemed complex at first but became easier once we factored and canceled wherever possible. Simplification helps us eliminate unnecessary complexity and reveals the underlying structure of the expression. This can often lead to easier limit evaluation or finding where a function might be undefined.

Pay special attention to:

• Combining like terms
• Cancelling common factors
• Rewriting expressions in equivalent forms
  

In this context, it's about transforming \( 2x^2 - x \) into \( x(2x - 1) \) to facilitate the cancellation of \( x \) in the numerator and the denominator. This step was crucial in simplifying our expression to a form where the limit could be evaluated quickly and accurately.

Factoring Polynomials

Factoring polynomials means writing a polynomial as a product of its factors, which are simpler expressions that can be multiplied to obtain the original polynomial. This technique is extremely useful in simplifying mathematical expressions and functions.

In the given exercise, we needed to simplify \( 2x^2 - x \). To do this, we factored the polynomial expression by identifying the common factor, which in this case is \( x \). This transforms the expression into \( x(2x - 1) \).

Key steps in factoring polynomials include:

• Looking for a common factor in all terms
• Applying the distributive property in reverse
• Breaking down complex polynomials into simpler (often binomial or trinomial) factors
  

Factoring not only simplified the initial expression, it allowed us to cancel the common \( x \), making it possible to directly evaluate the expression as \( x \) approached zero. This insight is vital in transforming a problem that seems complex into one that is easily solvable.