Problem 25 - Precalculus: Mathematics for Calculus

Question

Finding Limits Evaluate the limit if it exists. $$\lim _{h \rightarrow 0} \frac{(2+h)^{2}-4}{h}$$

Step-by-Step Solution

Answer

The limit is 4.

Step 1: Understand the Expression

We are given the limit \( \lim _{h \rightarrow 0} \frac{(2+h)^{2}-4}{h} \). Our goal is to find this limit as \( h \) approaches zero. The expression inside the limit is a quotient, and direct substitution of \( h = 0 \) would result in division by zero. Hence, some algebraic manipulation is required.

Step 2: Expand the Numerator

The numerator \((2+h)^2 - 4\) can be expanded. Use the formula \((a+b)^2 = a^2 + 2ab + b^2\):\[(2+h)^2 = 4 + 4h + h^2\]Thus, our numerator becomes:\[4 + 4h + h^2 - 4 = 4h + h^2\]

Step 3: Simplify the Fraction

The expression is now \( \frac{4h + h^2}{h} \). Factor out an \( h \) from the numerator:\[\frac{h(4 + h)}{h}\]Cancel the \( h \) in the numerator and denominator:\[4 + h\]

Step 4: Evaluate the Limit

Substitute \( h = 0 \) into the simplified expression:\[\lim _{h \rightarrow 0} (4 + h) = 4 + 0 = 4\]Therefore, the limit of the original expression as \( h \) approaches 0 is 4.

Key Concepts

Limits

Limits are a fundamental concept in calculus, serving as the foundation for understanding how functions behave as they approach a certain point. They explain what the value of a function is as it gets very close to a specific input. In our example, we want to determine what the expression \( \lim _{h \rightarrow 0} \frac{(2+h)^{2}-4}{h} \) approaches as \( h \) goes to zero.

To find a limit, you check what happens as the variable gets closer and closer to a given point. Here, by substituting very small values of \( h \) close to zero into the function, the aim is to figure out the value the expression is converging to.

• Direct substitution - Plug the value into the function if it doesn't result in undefined actions like division by zero.
• Algebraic manipulation - Rearrange the expression so direct substitution becomes possible.

For this exercise, immediate substitution isn’t viable as it results in division by zero, hence necessitating further manipulation.

Algebraic Manipulation

Algebraic manipulation is the process of transforming an expression by applying algebraic rules to simplify or rearrange it. This is important in limits when direct substitution leads to undefined results such as division by zero. In our specific example, the expression \( \lim _{h \rightarrow 0} \frac{(2+h)^{2}-4}{h} \) requires manipulation.

The initial step involves expanding \((2+h)^2 - 4\) by using the expansion of binomials. This means applying the formula \((a+b)^2 = a^2 + 2ab + b^2\), where \(2\) and \(h\) take the places of \(a\) and \(b\).

• By expanding \((2+h)^2\), you derive \(4 + 4h + h^2\).
• Subtract \(4\), resulting in \(4h + h^2\).

This simplifies the numerator to be factored and written as \(h(4 + h)\), which then allows for canceling out \(h\) from the numerator and denominator.

Substitution Method

The substitution method in limits involves replacing the variable with a specific value that the variable is approaching. This step is usually conducted after algebraic manipulation has simplified the expression enough to prevent undefined operations. After manipulating the expression \( \lim _{h \rightarrow 0} \frac{4h + h^2}{h} \) to \(4 + h\), substitution becomes straightforward.

With the expression now simplified, substitute \( h = 0 \) into \( 4 + h \) to evaluate the limit:

• Replace \( h \) with \(0\).
• Calculate the result, yielding \(4 + 0 = 4\).

This demonstrates that as \( h \) approaches zero, the expression's value approaches \(4\). Substitution is often the final step, offering a precise understanding of the function's behavior at the specified limit point.