Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 7-18, find the indicated limit. In most cases, it will be wise to do some algebra first (see Example 2). $$ \lim _{h \rightarrow 0} \frac{(x+h)^{2}-x^{2}}{h} $$

Step-by-Step Solution

Answer

The limit is \(2x\).

Step 1: Understanding the Limit

The problem asks to find the limit of a function as the variable \(h\) approaches 0. The expression given is \( \lim _{h \rightarrow 0} \frac{(x+h)^{2}-x^{2}}{h} \). This is a difference quotient that resembles the definition of the derivative of a function.

Step 2: Expanding the Numerator

First, expand the expression \((x+h)^2 - x^2\). Using the distributive property (\(a+b)^2 = a^2 + 2ab + b^2\), we have: \[(x+h)^2 = x^2 + 2xh + h^2\]. Substitute this back into the numerator: \[(x^2 + 2xh + h^2) - x^2\].

Step 3: Simplifying the Expression

After substitution, simplify the expression by canceling out \(x^2\): \[(x^2 + 2xh + h^2) - x^2 = 2xh + h^2\]. Thus, the expression becomes \(\frac{2xh + h^2}{h}\).

Step 4: Factoring and Cancelling

Factor \(h\) from the new numerator: \[h(2x + h)\]. This allows for simplification by cancelling \(h\) in the numerator and denominator: \[\frac{h(2x + h)}{h} = 2x + h\].

Step 5: Evaluating the Limit

Now, evaluate the limit as \(h\) approaches 0: \[\lim_{h \rightarrow 0} (2x + h)\]. Since \(h\) is approaching 0, substitute \(h = 0\) into the expression: \[2x + 0 = 2x\].

Key Concepts

Difference Quotient

The difference quotient is a fundamental concept in calculus that offers a way to measure how a function changes as its input changes. The typical form of a difference quotient is \( \frac{f(x+h)-f(x)}{h} \), where \( h \) is a small increment in \( x \). It essentially calculates the average rate of change of the function over this small interval.
Understanding this concept is crucial, as it paves the way for finding derivatives.In the exercise provided, the expression \( \frac{(x+h)^{2}-x^{2}}{h} \) represents a difference quotient. Here, \( (x+h)^{2} \) and \( x^{2} \) are specific instances of the function \( f(x) \). As \( h \) approaches zero, this quotient gives us information about the function's derivative at a point.

Algebraic Simplification

Algebraic simplification is the process of breaking down complex algebraic expressions into simpler, more manipulable forms. Before evaluating limits, it's often necessary to simplify the given expression using algebraic techniques.
In this task, the expression \( (x+h)^{2}-x^{2} \) is expanded using the distributive property:

• \( (x+h)^{2} = x^{2} + 2xh + h^{2} \).

When simplified, this becomes \( 2xh + h^{2} \) by canceling out \( x^2 \) from the positive and negative terms. This simplification is essential as it prepares the expression for the next step of the procedure.

Derivative

The derivative is a central concept in calculus that represents an instantaneous rate of change of a function with respect to one of its variables. It can be viewed as the slope of the tangent line to the function's curve at a given point.
The difference quotient that we worked with earlier forms the basis of the derivative's definition:

• The derivative of a function \( f(x) \) at a point \( x \) is defined as \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \).

In the example, as \( h \) approaches zero, the expression \( \frac{(x+h)^2 - x^2}{h} \) simplifies to \( 2x + h \). Further evaluation delivers \( 2x \), marking the derivative of \( x^2 \). This outcome illustrates the derivative's role in measuring how \( x^2 \) changes instantaneously.

Limit Evaluation

Limit evaluation is a technique used to determine the behavior of a function as the input approaches a specific value. In calculus, limits are essential for defining derivatives and integrals.
In the given solution, the goal was to evaluate the limit of the expression \( \lim _{h \rightarrow 0} (2x + h) \).

• Since \( h \) is approaching zero, substitute \( h = 0 \) into the simplified expression \( 2x + h \).

Doing so results in \( 2x + 0 = 2x \). This straightforward substitution shows that the final calculated limit is \( 2x \), confirming that all algebraic manipulations and limits align perfectly.
Understanding how to evaluate limits correctly is key to mastering calculus.