Evaluating Limits of Rational Functions

The question asks us to find the limit of the rational function \(\frac{x^3 + x^2}{x^2 + 3x + 2}\) as \(x\) approaches 2. Evaluating limits is a fundamental concept in calculus.

Understanding the Limit Problem

We need to find the value that the function approaches as the input \(x\) gets closer and closer to 2, without necessarily being equal to 2. For rational functions (a polynomial divided by another polynomial), the first step in evaluating a limit as \(x\) approaches a finite number is usually direct substitution.

Step-by-Step Solution: Direct Substitution

Let's substitute \(x = 2\) directly into the function \(\frac{x^3 + x^2}{x^2 + 3x + 2}\).

Substitute \(x=2\) into the numerator:

Numerator = \(x^3 + x^2 = (2)^3 + (2)^2\)

\( = 8 + 4 = 12\)

Substitute \(x=2\) into the denominator:

Denominator = \(x^2 + 3x + 2 = (2)^2 + 3(2) + 2\)

\( = 4 + 6 + 2 = 12\)

Analyzing the Result of Substitution

After substituting \(x=2\), we get the form \(\frac{12}{12}\). Since the denominator is not zero and the result is a finite number (1), the limit as \(x\) approaches 2 is simply the value obtained through direct substitution.

\(\mathop {\lim }\limits_{x \to 2}\frac{x^3 + x^2}{x^2 + 3x + 2} = \frac{(2)^3 + (2)^2}{(2)^2 + 3(2) + 2} = \frac{8 + 4}{4 + 6 + 2} = \frac{12}{12} = 1\)

Conclusion on the Limit Value

The limit of the function \(\frac{x^3 + x^2}{x^2 + 3x + 2}\) as \(x\) approaches 2 is 1.

Revision Table: Key Concepts

Additional Information on Evaluating Limits

While direct substitution worked in this case, it's important to know other methods for evaluating limits, especially when direct substitution results in an indeterminate form like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).

• Factoring and Cancelling: If direct substitution gives \(\frac{0}{0}\) for a rational function, try factoring the numerator and denominator to cancel out common factors that cause the zero in the denominator.
• L'Hopital's Rule: For indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), L'Hopital's Rule states that \(\mathop {\lim }\limits_{x \to c}\frac{f(x)}{g(x)} = \mathop {\lim }\limits_{x \to c}\frac{f'(x)}{g'(x)}\), provided the limit on the right exists. This involves taking the derivative of the numerator and the derivative of the denominator separately.
• Graphing: Sometimes, sketching the graph of the function can help visualize the value the function approaches as \(x\) gets close to a specific point.
• Numerical Approach: Evaluating the function for values of \(x\) very close to the limit point (both slightly less and slightly more than the point) can help estimate the limit.

In our specific problem, the direct substitution method was sufficient because the denominator was non-zero at \(x=2\).