Problem 17 - 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). $$ \lim _{h \rightarrow 0} \frac{(2+h)^{2}-4}{h} $$

Step-by-Step Solution

Answer

The limit is 4.

Step 1: Expand the Expression

First, expand the expression in the numerator. We start with \[(2+h)^{2} = (2+h) \cdot (2+h) = 4 + 4h + h^{2}.\] Thus, the expression becomes \[\frac{(4 + 4h + h^2) - 4}{h}.\]

Step 2: Simplify the Numerator

Simplify the expression by eliminating like terms in the numerator. Subtract 4 from 4:\[4 + 4h + h^2 - 4 = 4h + h^2.\] The expression is now \[\frac{4h + h^2}{h}.\]

Step 3: Factor and Cancel Terms

Factor out an \(h\) from the terms in the numerator: \[h(4 + h)\].Next, cancel the \(h\) from the numerator and denominator:\[\frac{h(4 + h)}{h} = 4 + h.\]

Step 4: Evaluate the Limit

Substitute \(h = 0\) in the simplified expression:\[\lim_{h \rightarrow 0} (4 + h) = 4 + 0 = 4.\]

Key Concepts

Limit Evaluation

When we talk about limit evaluation in calculus, we are determining the value that a function approaches as the variable gets closer to a specific point. Limits are foundational in calculus and are necessary for defining derivatives and integrals.
In this exercise, we need to find the limit of a function as the variable \(h\) approaches 0:

• Begin with rewriting the original problem in a more approachable form. This simplifies the process and makes it possible to directly substitute when \(h\) tends to 0.
• Once simplified, evaluate the expression by plugging in the value \(h = 0\). This helps in understanding how the function behaves near that point.

Evaluating limits helps understand the behavior of functions at points of discontinuity and how they behave in their domain. This is particularly useful for identifying instantaneous rates of change.

Algebraic Simplification

Algebraic simplification is a key process when dealing with calculus problems, especially with limits. By simplifying algebraic expressions, you are reducing them to a more manageable form.
For this limit problem, algebraic simplification involves:

• First, expanding complex expressions, which is a typical first step, helps in isolating terms that may be canceled or simplified further.
• Then, reducing the expression by canceling out terms that are common in both the numerator and the denominator. In our example, simplify \((2+h)^2 \) to \(4 + 4h + h^2\).

By using algebraic simplification, we can often make problems much easier to handle, making subsequent steps, like factoring or limiting, more accessible and less error-prone.

Factoring Polynomials

Factoring polynomials is often a crucial step in simplifying expressions within calculus problems. This allows us to cancel terms out more efficiently, making it easier to evaluate limits.
To factor a polynomial, follow another straightforward process:

• Identify common factors in each term of the polynomial. In our exercise, factor out an \(h\) from \(4h + h^2\), which gives \(h(4 + h)\).
• Once factored, cancel any terms that appear in both the numerator and the denominator. This cancellation simplifies our expression significantly, often eliminating terms that cause indeterminate forms like \(\frac{0}{0}\).

Understanding factoring is vital because it simplifies the computation considerably and provides clarity in evaluating limits or finding derivatives.

Numerator Simplification

The process of numerator simplification is necessary for addressing calculus limit problems. It involves honing in on the numeric component of a fraction to ease solving the problem.
In this exercise:

• Once the expression in the numerator was expanded, we simplified it by removing redundant terms, for example, subtracting 4 from both sides of the numerator: \(4 + 4h + h^2 - 4 = 4h + h^2\).
• This simplification makes other steps, like factoring and cancelling, more straightforward and manageable, paving the way for an easy limit evaluation.

Numerator simplification ensures that the core components of an expression are clear and minimal, reducing errors in solving later analytical calculus processes. It's all about clearing the path for easier computation.