Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

, find the indicated limit. In most cases, it will be wise to do some algebra first. $$ \lim _{x \rightarrow 0} \frac{x^{4}+2 x^{3}-x^{2}}{x^{2}} $$

Step-by-Step Solution

Answer

The limit is -1.

Step 1: Simplify the Expression

Begin by simplifying the fraction \( \frac{x^{4}+2x^{3}-x^{2}}{x^{2}} \). Notice that each term in the numerator \( x^4 + 2x^3 - x^2 \) can be divided by \( x^2 \). Factor \( x^2 \) out of the numerator:\[\frac{x^2(x^2 + 2x - 1)}{x^2}\]This simplifies to \( x^2 + 2x - 1 \) after canceling \( x^2 \) from the numerator and denominator.

Step 2: Substitute the Limit Value

Now that we have simplified the expression to \( x^2 + 2x - 1 \), substitute \( x = 0 \) into the expression:\[(0)^2 + 2(0) - 1 = -1\]

Step 3: Calculate the Limit

After substitution, we find the expression evaluated to \(-1\), thus the limit is \(-1\). Therefore, \[\lim_{x \rightarrow 0} \frac{x^{4}+2x^{3}-x^{2}}{x^{2}} = -1\]

Key Concepts

Algebraic Simplification

Algebraic simplification helps to transform complex expressions into simpler forms. In this exercise, simplification is achieved by factoring and reducing the given expression using basic algebraic operations.

• Identify common factors in the numerator and the denominator.
• Factor out the common term, in this case, extracting \(x^2\) from the terms \(x^4 + 2x^3 - x^2\).
• This process simplifies the original algebraic fraction substantially. Here, \(x^2(x^2 + 2x - 1)/x^2\) becomes simply \(x^2 + 2x - 1\).

Algebraic simplification facilitates limit calculation by removing indeterminate forms or zero denominators. This is especially useful when substituting the limit value would otherwise result in an undefined expression. By transforming the expression, we make it easier to plug the value directly into the new expression.

Limit Calculation

Limit calculation is about finding the value that a function approaches as the variable gets closer to a specific point. In this problem, we are calculating the limit as \(x\) tends to 0.

• First, simplify the expression to make substitution easier, as we did in the previous section.
• Directly substitute \(x = 0\) into the simplified expression \(x^2 + 2x - 1\).
• This yields the result, \(-1\), indicating that as \(x\) approaches 0, the function approaches \(-1\).

Limit calculation can clarify how a function behaves near certain points, providing insights into continuity and the behavior of the function near the boundaries. This method combines intuitive substitution with careful simplification to avoid mistakes.

Polynomial Division

Polynomial division is similar to long division with numbers but involves dividing expressions containing variables instead. In this exercise, polynomial division simplifies the expression \(\frac{x^{4}+2x^{3}-x^{2}}{x^{2}}\). The steps are straightforward:

• Identify the polynomial expressions in the numerator and denominator.
• Factor out the greatest common divisor, here \(x^2\), from the numerator \(x^4 + 2x^3 - x^2\).
• After factoring, the expression becomes \(x^2(x^2 + 2x - 1)/x^2\).

Canceling the common factor \(x^2\) leaves us with a simpler expression, \(x^2 + 2x - 1\), which is much easier to evaluate. Polynomial division is crucial when dealing with limits as it helps eliminate complex terms, making it easier to find the behavior of a function as \(x\) approaches a given value.