Problem 52 - Precalculus

Question

Use a table of values to evaluate the following limits as \(x\) increases without bound. $$\lim _{x \rightarrow \infty} \frac{10-3 x^{2}}{10-3 x^{3}}$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Analyze the function

Consider the function \( f(x) = \frac{10 - 3x^2}{10 - 3x^3} \). As \( x \) increases without bound, both the numerator and the denominator become dominated by their highest power terms. The highest power of \( x \) in the numerator is \( x^2 \) and in the denominator is \( x^3 \).

Step 2: Simplify the fraction

To evaluate the limit, simplify the fraction by dividing both the numerator and the denominator by \( x^3 \), the highest power in the denominator: \( f(x) = \frac{\frac{10}{x^3} - \frac{3}{x}}{\frac{10}{x^3} - 3} \).

Step 3: Consider the limit of each term as x approaches infinity

As \( x \to \infty \), notice that \( \frac{10}{x^3} \to 0 \) and \( \frac{3}{x} \to 0 \). Thus, the function simplifies to \( f(x) \approx \frac{0 - 0}{0 - 3} = \frac{0}{-3} \).

Step 4: Evaluate the limit

The limit of the function as \( x \to \infty \) is \( \frac{0}{-3} = 0 \). Therefore, \( \lim_{x \rightarrow \infty} \frac{10-3 x^{2}}{10-3 x^{3}} = 0 \).

Key Concepts

Table of Values

A table of values is a helpful tool in understanding how a function behaves as the input values change. Particularly when dealing with limits, a table may illustrate the trend of the function as it approaches a certain point or as it extends to infinity.
Using a table involves choosing values for \( x \) that grow progressively larger and then calculating the corresponding \( f(x) \).
For example, consider the function: \( f(x) = \frac{10 - 3x^2}{10 - 3x^3} \).

• When \( x = 1 \), \( f(1) = \frac{10 - 3 \cdot 1^2}{10 - 3 \cdot 1^3} = \frac{7}{7} \).
• As \( x = 10 \), \( f(10) = \frac{10 - 3 \cdot 10^2}{10 - 3 \cdot 10^3} = \frac{-290}{-2990} \).
• As \( x \) becomes very large, both the numerator and the denominator are dominated by the higher power terms, \( x^2 \) and \( x^3 \) respectively.

Seeing the values in tabulated form paints a clear picture of how \( f(x) \) approaches zero as we move towards infinity.

Dominant Term

In calculus, finding the dominant term is a crucial step when evaluating limits, especially as variables approach infinity.
The dominant term in a polynomial is the term with the greatest exponent of \( x \).
In our function \( f(x) = \frac{10 - 3x^2}{10 - 3x^3} \), we have:

• Numerator: Dominant term is \( -3x^2 \)
• Denominator: Dominant term is \( -3x^3 \)

These dominant terms largely define the behavior of the function as \( x \) grows larger. Recognizing these allows us to focus on significant contributors to the limit computation.The dominance of \( x^3 \) in the denominator, in particular, suggests more power than \( x^2 \) in the numerator, which hints at the result approaching zero.

Infinity

Infinity in calculus often represents a concept rather than a concrete number. We encounter limits extending to infinity either by inputs or outputs.
When analyzing limits as \( x \) approaches infinity, it's about understanding what happens to the function's value as \( x \) becomes exceedingly large.In dealing with our function, \( \lim_{x \to \infty} \frac{10 - 3x^2}{10 - 3x^3} \), we observe:

• The terms affected by multiplication or division by \( x \) head towards zero because the infinitesimal nature of \( \frac{1}{x} \) becomes pronounced.
• Both \( \frac{10}{x^3} \) and \( \frac{3}{x} \) vanish, simplifying the expression to primarily notice the behavior driven by constant and higher power terms.

Recognizing this helps ascertain that \( f(x) \) simplifies to \( \frac{0}{-3} = 0 \) when \( x \) is infinitely large.

Simplifying Fractions

Simplifying fractions facilitates easier calculation and understanding, especially when addressing limits. Here, we practice simplifying by factoring and reducing unnecessary complexity in expressions.
In the function \( f(x) = \frac{10 - 3x^2}{10 - 3x^3} \), simplifying involves:

• Divide each term by the highest power present in the denominator, \( x^3 \).
• This results in \( f(x) = \frac{\frac{10}{x^3} - \frac{3}{x}}{\frac{10}{x^3} - 3} \).
• As \( x \to \infty \), terms like \( \frac{10}{x^3} \) and \( \frac{3}{x} \) reduce to zero.

By focusing on these reductions, we effectively grasp how the dominant terms guide behavior as larger terms diminish, confirming that \( \lim_{x \to \infty} = 0 \).