Problem 30 - Calculus

Question

Use algebra to evaluate the limit. $$\lim _{h \rightarrow 0} \frac{(2+h)^{2}-4}{h}$$

Step-by-Step Solution

Answer

The limit is 4.

Step 1: Understand the Limit Expression

The given limit expression is \( \lim _{h \rightarrow 0} \frac{(2+h)^{2}-4}{h} \). This is an indeterminate form because direct substitution of \( h = 0 \) results in \( \frac{0}{0} \). We need to simplify the expression to resolve this indeterminate form.

Step 2: Expand the Binomial

Expand the term \( (2 + h)^2 \).\[ (2 + h)^2 = 4 + 4h + h^2 \]

Step 3: Substitute the Expansion

Substitute \( 4 + 4h + h^2 \) back into the limit expression:\[ \lim _{h \rightarrow 0} \frac{(4 + 4h + h^2) - 4}{h} \]

Step 4: Simplify the Expression

Simplify the numerator:\[ (4 + 4h + h^2) - 4 = 4h + h^2 \]Thus, the expression becomes:\[ \lim _{h \rightarrow 0} \frac{4h + h^2}{h} \]

Step 5: Factor and Cancel Terms

Factor out \( h \) from the numerator:\[ \frac{h(4 + h)}{h} \]Cancel the \( h \) terms:\[ \lim _{h \rightarrow 0} (4 + h) \]

Step 6: Evaluate the Limit

Since \( h \) approaches 0, substitute 0 into the simplified expression:\[ 4 + 0 = 4 \] Thus, \( \lim _{h \rightarrow 0} (4 + h) = 4 \).

Key Concepts

Indeterminate Forms

Indeterminate forms in limits occur when a mathematical expression is not immediately solvable. The classic indeterminate form is \( \frac{0}{0} \), which can happen when directly substituting a value into a limit. This often signals that the equation needs simplification for proper evaluation. In this exercise, substituting \( h = 0 \) into the expression \( \lim_{h \rightarrow 0} \frac{(2+h)^{2}-4}{h} \) results in \( \frac{0}{0} \), an indeterminate form. Indeterminate responses challenge mathematicians because they are forms that require more work to be resolved. This is typically done by rewriting the expression. Once the expression is simplified, you can then evaluate the limit straightforwardly. Recognizing when an indeterminate form occurs is crucial in the process of finding limits in calculus.

Binomial Expansion

Binomial expansion is a method of expanding expressions raised to a power that involves two terms, such as \((a + b)^n\). This expansion follows a specific pattern based on the binomial theorem, allowing expressions like \((2 + h)^2\) to be expanded easily.For example, in the problem given, \((2 + h)^2\) needs expanding. By applying binomial expansion, the expression is rewritten as:

• \( (2 + h)^{2} = 4 + 4h + h^2 \)

This straightforward expansion simplifies the limit function, making further calculations easier. It involves recognizing and distributing each component of the binomial expression separately. Mastering this technique enables you to simplify expressions that would otherwise be difficult to manage.

Limit Evaluation

Evaluating limits involves simplifying an expression to substitute in the approach value. In this problem, having handled the indeterminate form and expanded the binomial, the goal is to simplify the expression enough to replace \( h \) with zero.Starting with the simplified expression \( \frac{4h + h^2}{h} \), the factoring step \( h(4 + h) \) allows us to cancel \( h \) from the denominator and numerator:

• \( \lim_{h \rightarrow 0} (4 + h) \)

Once this cancellation is made, substituting \( h = 0 \) becomes straightforward since the expression now does not lead to an undefined form. The result is \( 4 + 0 = 4 \). Being meticulous about each step in the limit evaluation process is essential for mathematical accuracy, particularly when starting from an indeterminate form.