Problem 35 - Calculus

Question

Explain what is wrong with the statement. since \(\lim _{h \rightarrow 0}(2+h)^{2}=4,\) we have $$\lim _{h \rightarrow 0} \frac{(2+h)^{2}-2^{2}}{h}=0$$

Step-by-Step Solution

Answer

The statement incorrectly claims \(\lim_{h \rightarrow 0} \frac{(2+h)^{2}-2^{2}}{h}=0\); it is actually 4.

Step 1: Understand the Statement

The statement is about calculating two limits and concludes that one limit results from the other. It claims that since the limit \(\lim_{h \rightarrow 0}(2+h)^{2}=4\), then another limit, \(\lim_{h \rightarrow 0} \frac{(2+h)^{2}-2^{2}}{h}=0\), must also be true.

Step 2: Evaluate the First Limit

Let's evaluate the given first limit: \(\lim_{h \rightarrow 0}(2 + h)^2 = 4\). As \(h\) approaches 0, \((2+h)^2\) approaches \(4\), which is correctly stated since \((2+0)^2 = 4\).

Step 3: Analyze the Second Limit Expression

The second limit involves the expression \(\frac{(2+h)^{2} - 2^{2}}{h}\). This represents the derivative of \((2+h)^2\) at \(h=0\), which is correctly expressed as a limit.

Step 4: Expand and Simplify the Numerator

Expand \((2+h)^2\) to get \(4 + 4h + h^2\). The expression becomes \((4 + 4h + h^2 - 4) = 4h + h^2\).

Step 5: Factor and Simplify the Fraction

Substitute the expanded numerator back: \(\frac{4h + h^2}{h}\). Factor out \(h\): \(h(4+h)\). Simplifying, we get \(\frac{4h + h^2}{h} = 4 + h\).

Step 6: Evaluate the Simplified Limit

Now calculate \(\lim_{h \rightarrow 0} (4 + h)\). As \(h\) approaches 0, the limit is \(4\), not 0.

Step 7: Conclusion

The statement claims the incorrect limit. The actual limit \(\lim_{h \rightarrow 0} \frac{(2+h)^{2}-2^{2}}{h}\) equals to \(4\) rather than 0. The original reasoning was flawed.

Key Concepts

Limits

Limits are a foundational concept in calculus, playing a crucial role in the study of derivatives and integrals. In essence, a limit describes the value that a function approaches as the input approaches a certain point. When dealing with limits, it is important to understand that we analyze the behavior of the function as it gets very close to a particular point, even if it never actually reaches that point.
For example, the limit \( \lim_{h \rightarrow 0}(2+h)^2 = 4\) demonstrates that as \(h\) gets closer and closer to zero, the expression \((2+h)^2\) becomes increasingly close to \(4\). It's essential to follow the function's behavior rather than just substituting the value directly, particularly when evaluating more complex expressions.

• Always consider the approaching behavior of the variable.
• Replace "\(h = 0\)" with the concept of \("h\) approaching 0".
• Avoid discontinuities or undefined expressions by applying the limit process.

Understanding limits helps build the foundation for discussing other key calculus concepts such as continuity and differentiability.

Simplification

Simplification is the process of making an expression easier to work with or understand. When dealing with limit problems, simplification ensures that we can accurately determine the value of a limit by removing any ambiguities or complexities in expressions.
For example, in the expression \((2+h)^2 - 2^2\), we expand to \((4 + 4h + h^2 - 4) = 4h + h^2\). Through simplification, we follow these steps:

• Expand complex expressions to simpler form.
• Factor out common terms to make calculations manageable.

This allows us to consider the behavior of the expression at \(h = 0\), ensuring an accurate calculation of the limit. In our case, by factoring \(h\) out, we simplify \(\frac{4h + h^2}{h}\) into \(4 + h\), which makes it easier to evaluate as \(h\) approaches zero. Therefore, simplification is an indispensable step in ensuring precise limit calculations.

Polynomial Expansion

Polynomial expansion involves rewriting a polynomial or expression in an expanded form. This process is critical when dealing with derivatives and limits, as it allows for simpler manipulation and understanding of the expression's behavior.
Consider the expression \((2+h)^2\). When expanded, it translates into \(4 + 4h + h^2\), which allows us to better visualize and understand each component of the expression. Here are key points to remember about polynomial expansion:

• Expand each term in a polynomial expression individually.
• Ensure that all distributed multiplications are correctly executed.

By expanding \((2+h)^2\), it becomes simpler to determine its derivative operations or limit simplifications, because each term is laid out clearly. This straightforward expansion prevents mistakes in operations and aids in an accurate understanding of limit problems. With polynomial expansion, it's not just about rewriting an expression; it's about giving you the tools you need to handle complex calculus challenges effectively.