Problem 25 - APEX Calculus

Question

Evaluate the given limit. $$ \lim _{x \rightarrow \infty} \frac{x^{3}+2 x^{2}+1}{x-5} $$

Step-by-Step Solution

Answer

The limit is \( \infty \).

Step 1: Identify the Dominant Terms

In the expression \( \frac{x^3+2x^2+1}{x-5} \), the dominant terms as \( x \rightarrow \infty \) are \( x^3 \) in the numerator and \( x \) in the denominator. These terms grow faster than the others, so they will determine the behavior of the limit.

Step 2: Simplify the Expression

To simplify \( \frac{x^3+2x^2+1}{x-5} \), divide every term in the numerator and the denominator by \( x \), which is the highest power in the denominator. This gives us \( \frac{x^2 + \frac{2}{x}x + \frac{1}{x^2}}{1 - \frac{5}{x}} \).

Step 3: Evaluate the Limit

As \( x \rightarrow \infty \), the terms \( \frac{2}{x} \), \( \frac{1}{x^2} \), and \( \frac{5}{x} \) all approach 0. Therefore, the expression simplifies to \( \frac{x^2 + 0 + 0}{1 - 0} = x^2 \).

Step 4: Determine the Limit's Behavior

The result \( x^2 \) indicates that the limit grows without bound as \( x \rightarrow \infty \). Therefore, the limit does not approach a finite number.

Key Concepts

Dominant Terms

When working with limits at infinity, identifying dominant terms becomes crucial. We use dominant terms to predict how functions behave as variables grow very large. In the expression \( \frac{x^3 + 2x^2 + 1}{x - 5} \), we are interested in which terms in the numerator and denominator increase at the fastest rate as \( x \to \infty \).

In our example:

• Numerator: The term \( x^3 \) grows much faster than the terms \( 2x^2 \) and \( 1 \) as \( x \) increases.
• Denominator: The term \( x \) is the only term in the denominator, making it dominant by default.

These dominant terms dictate the behavior of the entire expression, thus simplifying our task when evaluating limits.

Simplifying Expressions

Once dominant terms are identified, the next step is simplifying the expression to make it easier to evaluate the limit. We do this by dividing each term by the highest power of \( x \) present in the denominator.

In the given expression \( \frac{x^3 + 2x^2 + 1}{x - 5} \), we divide each term by \( x \):

• Numerator becomes \( x^2 + \frac{2}{x}x + \frac{1}{x^2} \).
• Denominator becomes \( 1 - \frac{5}{x} \).

This method often transforms the original complex expression into a simpler form, where some terms may become negligible as \( x \to \infty \). Here, the terms such as \( \frac{2}{x} \), \( \frac{1}{x^2} \), and \( \frac{5}{x} \) approach zero, further simplifying the expression.

Behavior of Limits

The ultimate goal is understanding how an expression behaves as \( x \to \infty \). After simplifying the expression in our problem, it becomes \( \frac{x^2 + 0 + 0}{1 - 0} \), which simplifies to \( x^2 \).

As \( x \) increases, \( x^2 \) expands without bounds, indicating that the limit of the expression is infinite.

Key insights about behavior of limits:

• If, after simplification, the expression tends to a constant, the limit is that constant.
• If it contains terms that grow endlessly, like \( x^2 \), the limit is infinite.
• If it oscillates without settling towards a single value, the limit does not exist.

Hence, understanding the behavior of limits helps us categorize them more effectively, guiding how we interpret expressions as variables approach infinity.