Problem 32 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

The limit is \( x^2 \).

Step 1: Simplify the Expression

First, consider the given expression \( \frac{x^{2} h - x h^{2} + h^{3}}{h} \). Since the denominator is \( h \), we can simplify by dividing each term in the numerator by \( h \). This results in: \[ x^2 - xh + h^2. \]

Step 2: Apply the Limit

Now apply the limit as \( h \) approaches 0. Substitute \( h = 0 \) into the simplified expression: \[ x^2 - x(0) + 0^2 = x^2. \]

Step 3: Conclude the Limit

Conclude that the limit of the expression as \( h \) approaches 0 is \( x^2 \). This is because each term involving \( h \) in the simplified expression becomes 0.

Key Concepts

Limit Evaluation

When it comes to evaluating a limit, the focus is on determining the value that a function approaches as the variable gets closer to a specific point. In this exercise, you are asked to find the limit of a fraction as the variable \( h \) approaches zero. Evaluating limits provides a window into understanding the behavior of functions near specific points. The initial expression given is \( \lim_{h \to 0} \frac{x^2 h - x h^2 + h^3}{h} \). This expression might seem complex because having \( h \) both in the numerator and the denominator can lead to an undefined condition if simply substituted directly. To properly evaluate this limit, critical observation of the expression is necessary, ensuring that the expression is in a form that can be computed effectively.

Limit Simplification

Simplification is a crucial step when dealing with limits, especially when the expression might lead to an indeterminate form like \( \frac{0}{0} \). In this particular problem, the expression \( \frac{x^2 h - x h^2 + h^3}{h} \) can be simplified by dividing each term in the numerator by the common denominator \( h \). This is possible because \( h eq 0 \) when making simplifications; only at the very end, as part of the limit evaluation, do we consider \( h \to 0 \).After performing the division, the expression simplifies to \( x^2 - xh + h^2 \). This simplification is not only helpful but necessary to avoid dividing by zero and effectively evaluate the limit. Additionally, it reduces complexity, making it easier to apply other calculus concepts.

Limit Calculation Steps

The step-by-step calculation of a limit involves both simplifying the expression and applying limit properties. Let's break this down:**Step 1:** Simplifying the expression. We started with \( \frac{x^2 h - x h^2 + h^3}{h} \). By dividing each of the terms by \( h \), simplicity was achieved: the expression is reduced to \( x^2 - xh + h^2 \).**Step 2:** Applying the limit. With the simplified expression, we substitute \( h = 0 \), keeping in mind that only terms involving \( h \) will vanish because as \( h \to 0 \), terms like \( h \) and \( h^2 \) become zero. Applying the limit results in \( x^2 - x(0) + 0^2 = x^2 \).**Step 3:** Concluding the limit. The final step is straightforward; as all terms containing \( h \) disappear, we are left with \( x^2 \). This confirms the limit of the expression as \( h \) approaches zero. By following these straightforward steps systematically, limits become much easier and more manageable to compute.