Problem 25 - Calculus Early Transcendentals

Question

Determining limits analytically Determine the following limits or state that they do not exist. $$\lim _{x \rightarrow 0} \frac{x^{3}-5 x^{2}}{x^{2}}$$

Step-by-Step Solution

Answer

Answer: The limit of the expression is \(-5\).

Step 1: Factor out the common term in the numerator

The given expression is: $$\lim _{x \rightarrow 0} \frac{x^{3}-5 x^{2}}{x^{2}}$$ Let's factor out the common term (\(x^{2}\)) in the numerator: $$\lim _{x \rightarrow 0} \frac{x^2(x-5)}{x^{2}}$$

Step 2: Simplify the expression by cancelling the common term

Now, we can simplify the expression by cancelling out the common term, which is \(x^2\) in this case: $$\lim _{x \rightarrow 0} (x-5)$$

Step 3: Evaluate the limit

As the expression is now simplified, we can evaluate the limit by plugging in \(x=0\): $$\lim_{x \rightarrow 0} (x - 5) = (0 - 5) = -5$$ So, the limit of the given expression as \(x\) approaches \(0\) is \(-5\).

Key Concepts

Analytical Limit Determination

Analytical limit determination is a crucial concept in calculus. It involves finding the value that a function approaches as the input approaches a certain point. For the exercise provided, the task is to evaluate the limit of \( \lim_{x \rightarrow 0} \frac{x^3 - 5x^2}{x^2} \). The aim is to find what value the expression tends to as the variable \( x \) approaches zero.

Here is how you can think about analytical limit determination:

• Recognize the structure of the mathematical expression you are dealing with, like a fraction in this case.
• Before evaluating directly, determine if simplifying the expression might prevent the occurrence of undefined behavior (like dividing by zero).
• Finally, substitute the limiting value into the simplified expression to determine the actual limit.

This process helps avoid indeterminate forms, ensuring that the limit remains well-defined.

Factoring in Limits

Factoring is an essential step when dealing with limits, especially with polynomial expressions. In our exercise, the numerator \( x^3 - 5x^2 \) needs to be simplified. Factoring involves finding common terms or expressions within the function that can be simplified.

Let's break down factoring:

• Identify common factors: In this example, \( x^2 \) is common in the terms of the numerator.
• Factor it out: Rewrite the expression by factoring out \( x^2 \, \), obtaining \( x^2(x - 5) \).
• This reveals a structure you can simplify further by canceling common terms in the numerator and the denominator, leading to a simpler expression \((x - 5)\).

Using factoring helps manage complex expressions, making it easier to proceed to the next steps of determining the limit.

Simplifying Expressions in Calculus

Simplifying expressions is all about making a function easier to work with by reducing it to its most basic form. After factoring, as seen in the previous step, we achieved a much simpler expression: \( (x - 5) \).

Consider these aspects when simplifying for limits:

• Cancel identical terms in the numerator and the denominator, as we did with \( x^2 \) in this exercise. This step is crucial since it avoids division by zero or other undefined operations.
• Simplified expressions are often easier to evaluate, especially when substituting limits directly.
• If you're left with a linear or simple polynomial, as we are here, you directly substitute the approaching value (\( x = 0 \)) to find the limit without any further complicated operations.

Simplifying paves the way to evaluating limits without confusion, turning potentially complicated expressions into manageable problems.