Question

Find the limits of the following: \(\lim _{x \rightarrow \infty} \frac{3 x^{2}}{5 x+8}\)

Short answer

Answer: The limit does not exist.

Step-by-step solution

Identify the dominant terms

When x becomes very large, the dominant terms are the ones with the highest powers of x. In this case, in the numerator, the dominant term is \(3x^2\). In the denominator, the dominant term is \(5x\).

Divide both the numerator and denominator by \(x^2\)

To simplify the fraction, we can divide both the numerator and denominator by the highest power of x. In this case, that is \(x^2\). \(\frac{3x^2}{5x + 8} = \frac{\frac{3x^2}{x^2}}{\frac{5x + 8}{x^2}}\) This simplifies to: \(\frac{3}{\frac{5}{x} + \frac{8}{x^2}}\)

Evaluate the limit as x approaches infinity

Now we examine the behavior of the function as x becomes very large. As x approaches infinity, the terms \(\frac{5}{x}\) and \(\frac{8}{x^2}\) will both approach 0, because the denominators will become much larger than the numerators. Therefore, we have: \(\lim_{x \rightarrow \infty} \frac{3}{\frac{5}{x} + \frac{8}{x^2}} = \frac{3}{0 + 0} = \frac{3}{0}\) However, a limit cannot be found if the denominator approaches 0, which means the function is not defined at infinity. So, the limit of the function as x approaches infinity does not exist.

Key concepts

Dominant Terms

In calculus, when dealing with functions, especially as they approach infinity, it's crucial to identify the dominant terms. Dominant terms are the parts of a function that have the highest power of the variable, which significantly affects the behavior of the function at large values. To simplify:

• In the expression \(3x^2\), \(x^2\) is the dominant term because it has the highest exponent.
• In \(5x + 8\), \(5x\) is the dominant term for the same reason.

Recognizing these terms helps in simplifying complex rational expressions. For example, in the limit \(\lim_{x \rightarrow \infty} \frac{3x^2}{5x+8}\), identifying \(3x^2\) and \(5x\) as dominant terms allows us to effectively simplify the evaluation by focusing on the most influential components of each polynomial.

Infinite Limits

Infinite limits are an essential concept in calculus, particularly when analyzing the behavior of functions as a variable grows without bound. When \(x\) approaches infinity, we try to determine how a function behaves and whether it settles to a specific value or does not exist.In the example \(\lim_{x \rightarrow \infty} \frac{3}{\frac{5}{x} + \frac{8}{x^2}}\), both \(\frac{5}{x}\) and \(\frac{8}{x^2}\) shrink towards zero as \(x\) increases. This is because:

• The term \(\frac{1}{x}\) becomes minuscule, pushing the entire expression to close in on zero.

This demonstrates that understanding how each part of the expression reacts as infinity is approached helps clarify why certain limits do not exist or settle at a particular finite number. Here, however, the result "3 divided by 0" shows the function behaves unpredictably, signaling the limit does not exist.

Rational Functions

Rational functions are expressions formed by dividing one polynomial by another. They are typically written as \( \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials. These functions can have varied behaviors, especially at limits approaching infinity or near undefined points in their domain.To explore further:

• Understanding rational functions involves identifying where they are undefined due to zeros in the denominator.
• In the limit process, simplifying by dividing through by the highest power of \(x\) ensures you focus purely on dominant terms.

The example \(\frac{3x^2}{5x+8}\) is simplified by recognizing its core components—dominant terms—and how they inform behavior as \(x\) increases. This separation of the dominant elements allows us to make clear predictions regarding infinite limits, consistency in form, and where rational functions do not achieve a finite limit, as demonstrated when dividing by zero.