Problem 31 - Brief Applied Calculus

Question

Find the following limits without using a graphing calculator or making tables. $$ \lim _{h \rightarrow 0} \frac{4 x^{2} h+x h^{2}-h^{3}}{h} $$

Step-by-Step Solution

Answer

The limit is \( 4x^2 \).

Step 1: Recognize the Limit Form

We are asked to find the limit \( \lim_{h \to 0} \frac{4x^2 h + xh^2 - h^3}{h} \). This expression is of the indeterminate form \( \frac{0}{0} \) when \( h \to 0 \). Thus, we need to simplify to remove \( h \) from the denominator.

Step 2: Factor the Numerator

Observe that every term in the numerator \( 4x^2 h + xh^2 - h^3 \) has an \( h \) factor. We can factor \( h \) out: \( h(4x^2 + xh - h^2) \). Rewrite the original expression as \( \lim_{h \to 0} \frac{h(4x^2 + xh - h^2)}{h} \).

Step 3: Cancel the Common Factor

Since \( h eq 0 \) for the limit process, we can cancel \( h \) in the numerator and the denominator. The expression simplifies to \( \lim_{h \to 0} (4x^2 + xh - h^2) \).

Step 4: Evaluate the Limit

Now substitute \( h = 0 \) in the simplified expression: \( 4x^2 + x \cdot 0 - 0^2 = 4x^2 \). Since substitution does not lead to an indeterminate form, this is the limit.

Key Concepts

Indeterminate Forms

In calculus, indeterminate forms arise when attempting to evaluate limits that cannot be immediately determined. These forms, such as \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \), occur when direct substitution of the limit value into the denominator and numerator results in an undefined expression.
When encountering an indeterminate form, it signals that further algebraic manipulation or alternative methods are necessary to evaluate the limit. In the given exercise, substituting \( h = 0 \) results in the indeterminate form \( \frac{0}{0} \), indicating that simplification is essential to find a meaningful limit.
Recognizing indeterminate forms helps guide the choice of strategy to manage and appropriately resolve the limit question.

Algebraic Simplification

Algebraic simplification involves rewriting complex expressions into simpler, more manageable forms, often by factoring, expanding, or canceling terms. This technique is crucial for solving limit problems successfully. In our problem, each term in the numerator \( 4x^2 h + xh^2 - h^3 \) includes the variable \( h \), which allows \( h \) to be factored out.
By factoring out \( h \), the expression can be rewritten as \( h(4x^2 + xh - h^2) \). This step is key, as it prepares the expression for canceling with the \( h \) in the denominator, simplifying the entire problem.
Simplification not only removes indeterminate forms but also makes evaluating limits straightforward, avoiding the pitfalls of undefined expressions.

Limit Evaluation Steps

Limit evaluation is a systematic process where specific steps help to determine the outcome of a limit problem, especially when dealing with indeterminate forms.
Initially, recognize whether a limit results in an indeterminate form like \( \frac{0}{0} \). If it does, proceed with algebraic simplification. In the exercise, factoring out \( h \) from the numerator allows cancellation of \( h \) in the denominator, simplifying the problem.
After this simplification, substitute the limit value \( h = 0 \) into the reduced expression. The remaining terms produce a result that no longer presents an undefined form. Here, substituting into \( 4x^2 + xh - h^2 \) gives \( 4x^2 \), easily computing the limit.
Following these structured steps carefully solves the limit problem and overcomes the hurdles of indeterminate forms.