Problem 30 - Brief Applied Calculus

Question

Find the following limits without using a graphing calculator or making tables. $$ \lim _{h \rightarrow 0} \frac{5 x^{4} h-9 x h^{2}}{h} $$

Step-by-Step Solution

Answer

The limit is \( 5x^4 \).

Step 1: Understand the Limit Problem

We are asked to find the limit of the function \( \frac{5x^4h - 9xh^2}{h} \) as \( h \) approaches zero. This type of problem often involves simplifying the expression to remove \( h \) from the denominator.

Step 2: Factor Out \( h \) from the Numerator

The given expression is \( \frac{5x^4h - 9xh^2}{h} \). The numerator, \( 5x^4h - 9xh^2 \), can be factored by \( h \). Factoring \( h \) out gives us \( h(5x^4 - 9xh) \). The expression now becomes \( \frac{h(5x^4 - 9xh)}{h} \).

Step 3: Cancel \( h \) in the Expression

In the expression \( \frac{h(5x^4 - 9xh)}{h} \), the \( h \) in the numerator and the \( h \) in the denominator can be canceled out since \( h eq 0 \) (as \( h \) approaches 0 but is not zero). This simplifies the expression to \( 5x^4 - 9xh \).

Step 4: Evaluate the Limit as \( h \rightarrow 0 \)

Now that the expression is simplified to \( 5x^4 - 9xh \), substitute \( h = 0 \). This yields \( 5x^4 - 9x(0) = 5x^4 \).

Step 5: Conclude the Solution

With \( h \) approaching zero, the limit of the function is \( 5x^4 \).

Key Concepts

Limit Evaluation

In calculus, limit evaluation is the process of determining what value a function approaches as the input approaches a specified point. Limits are foundational for calculus, enabling the study of change and motion. Understanding limits involves considering what happens to a function as the input gets closer to a particular value. In our example, we evaluated the limit as \( h \) approaches 0. Often, the challenge is to simplify the expression to a form where this evaluation is straightforward, as not all expressions are directly evaluable due to undefined forms like division by zero. To successfully evaluate the limit:

• Simplify the function if it presents an indeterminate form, such as \( \frac{0}{0} \).
• Use algebraic techniques to rewrite the function.
• Substitute the limit point once simplification makes the expression evaluable.

These steps allow us to determine the behavior of functions at points that might seem inaccessible due to initial complexity.

Algebraic Simplification

Algebraic simplification makes complex expressions easier to handle, which is essential in calculus, especially for limits. In the given problem, we have the expression \( \frac{5x^4h - 9xh^2}{h} \). Directly substituting \( h = 0 \) leads to an undefined expression, so simplification is crucial.The simplification process involves:

• Identifying shared factors in expressions.
• Factoring out common terms to reduce the complexity.
• Rewriting expressions in an equivalent but simpler form.

In this case, we simplified the expression by factoring out \( h \) from the numerator. This approach reduced the complexity of the expression, allowing for easy evaluation of the limit. Algebraic simplification streamlines the process and reveals hidden structures that make limits and other calculus phenomena approachable.

Factorization

Factorization is a powerful tool in simplifying expressions for limit evaluation. It involves breaking down expressions into products of smaller or simpler expressions. This process helps reveal structures that facilitate further simplification, as seen in our problem.In the expression \( 5x^4 h - 9xh^2 \), both terms share a common factor of \( h \). Factoring out \( h \) gives us \( h(5x^4 - 9xh) \). This step is crucial because it cancels out the problematic \( h \) in the denominator. Factorization helps to manage terms that cause indeterminate forms like \( \frac{0}{0} \). By removing or simplifying these terms, the limit can be evaluated. This reflects the power of factorization in breaking down complex calculus problems into more manageable parts.

Variable Substitution

Variable substitution is a technique used to simplify expressions, making them easier to evaluate, especially when approaching a limit. After simplifying expressions through algebraic techniques and factorization, substitution gives a straightforward way to find the limit’s value. Once the expression \( 5x^4 - 9xh \) was achieved after simplification and factorization, the next step was variable substitution. We substituted \( h = 0 \), which resolved the expression to \( 5x^4 - 9x(0) = 5x^4 \). Substitution is powerful because it allows us to directly plug in the values needed to evaluate limits. It provides a clear picture of what happens to the function as the variable changes. When executed correctly, it leads to the conclusive evaluation of the limit, completing the problem-solving process in calculus.