Problem 21 - Brief Applied Calculus

Question

Find the following limits without using a graphing calculator or making tables. $$ \lim _{h \rightarrow 0}\left(5 x^{3}+2 x^{2} h-x h^{2}\right) $$

Step-by-Step Solution

Answer

The limit is \(5x^3\).

Step 1: Understanding the Limit

We need to find the limit of the expression \(5x^3 + 2x^2 h - xh^2\) as \(h\) approaches 0. This involves evaluating the behavior of the expression as \(h\) becomes very small.

Step 2: Separate Terms of the Expression

Consider the expression \(5x^3 + 2x^2 h - xh^2\). Notice that it is a polynomial and therefore, the limit can be determined by replacing \(h\) with 0 in the expression since polynomial functions are continuous.

Step 3: Evaluating Each Term Individually

Evaluate each term in the polynomial as \(h \rightarrow 0\).- The term \(5x^3\) is independent of \(h\), so it remains \(5x^3\).- The term \(2x^2h\) becomes \(2x^2 \cdot 0 = 0\) as \(h\) approaches 0.- The term \(xh^2\) becomes \(x \cdot 0^2 = 0\) as \(h\) approaches 0.

Step 4: Combine Results of Evaluated Terms

By evaluating the limits of each term and combining the results, the expression simplifies to: \[ 5x^3 + 0 + 0 = 5x^3 \]Thus, the limit of the original expression as \(h\) approaches 0 is \(5x^3\).

Key Concepts

Limit Evaluation

When we are evaluating the limit of a polynomial expression as a variable approaches a specific value, we are interested in what the expression looks like if we zoom in infinitely close to the point in question. In simple terms, this means observing the behavior of the function as a certain variable, in this case, \( h \), gets very close to zero.

• The goal is to pinpoint the expression's behavior and simplify it to deduce the ultimate value.
• For our expression \( 5x^3 + 2x^2 h - xh^2 \), this means breaking down each component to understand how they change as \( h \) approaches zero.

This step-by-step simplification helps clarify how each part of the expression contributes to the limit and aids in identifying any zeros, which are crucial when one or more terms vanish as a result of being multiplied by the approaching-zero variable.

Continuity of Polynomials

Polynomials are a class of functions in mathematics that have a very nice property called continuity. This means that they are smooth and unbroken; you can draw them without lifting your pen from the paper.

• Polynomials are continuous everywhere, meaning at every point and at every limit like \( h = 0 \) in our example.
• This allows us to substitute values directly into these expressions when finding limits, thereby simplifying the overall process.

In our exercise, because the expression is a polynomial, we can confidently replace \( h \) with 0 to get the limit directly as \( 5x^3 \). This reflects the seamless nature of polynomial functions, making evaluations of their limits straightforward.

Behavior as Variable Approaches Zero

The behavior of terms in an expression as a variable approaches zero determines how the whole expression behaves.

• Terms that are independent of this variable remain unaffected.
• Each term that includes the variable \( h \), such as \( 2x^2h \) and \( xh^2 \), diminishes or disappears completely.

When \( h \) approaches zero, we observe the dominance of terms not influenced by \( h \), such as \( 5x^3 \), which persists unaffected. This distinct behavior helps streamline the calculation by allowing terms that vanish to be ignored, focusing instead on those that remain impactful in producing the limit value of \( 5x^3 \). Thus, understanding the behavior as the variable shrinks to zero provides clear direction in determining the outcome of the limit.