Question

Find the limit. \(\lim _{x \rightarrow \infty} \frac{5 x^{3}+1}{10 x^{3}-3 x^{2}+7}\)

Short answer

The solution to \(\lim _{x \rightarrow \infty} \frac{5 x^{3}+1}{10 x^{3}-3 x^{2}+7}\) is \(\frac{1}{2}\)

Step-by-step solution

Identify highest degree terms

In both the numerator and the denominator, the term with the highest degree is \( x^{3} \). The coefficients of these terms are 5 in the numerator and 10 in the denominator.

Divide through by \( x^{3} \)

Divide every term in the numerator and the denominator by \( x^{3} \) to simplify the expression: \[\lim _{x \rightarrow \infty} \frac{5 + \frac{1}{x^{3}}}{10 - \frac{3}{x} + \frac{7}{x^{3}}}\].

Apply the limit

As \( x \rightarrow \infty \), the terms \(\frac{1}{x^{3}}\), \(\frac{3}{x}\), and \(\frac{7}{x^{3}}\) all approach 0. This gives: \[\lim _{x \rightarrow \infty} \frac{5 + 0}{10 - 0 + 0}\] which simplifies to \[\frac{5}{10}\].

Key concepts

Infinite Limits in Calculus

In calculus, when we talk about limits, we're exploring how a function behaves as its input approaches a certain value. One interesting scenario occurs when the input continues to grow larger and larger—towards infinity. This is what we call an **infinite limit**. Instead of focusing on a specific point, infinite limits regard the trend or direction a function is heading.

In our exercise, we want to find the limit of a rational function as \(x\) approaches infinity. It's crucial to observe how the terms in the equation behave as \(x\) becomes infinitely large. For instance, terms like \(\frac{1}{x}, \frac{1}{x^2}\) or \(\frac{1}{x^3}\) diminish to zero because division by a huge number shrinks these fractions significantly.

Understanding infinite limits helps in determining the behavior of functions at the edges of their domains. It allows mathematicians to predict long-term trends in various fields, from physics to economics.

Recognizing the Highest Degree Polynomial

The highest degree term in a polynomial is the term with the largest exponent on the variable. This term is especially important when evaluating limits of rational functions. In limits involving polynomial functions, the highest degree term dominates the behavior of the function as \(x\) approaches infinity.

In the original exercise, both the numerator and the denominator have their highest degree term as \(x^3\). The coefficient of the \(x^3\) term in the numerator is 5, while in the denominator it is 10. This is because the highest degree terms are the most significant ones when \(x\) grows very large, and lower degree terms become negligible.

By focusing on the highest degree terms, we simplify both the understanding and computation of infinite limits. It helps us streamline large expressions to something more manageable, which makes understanding the overall trend of the function much more straightforward.

Simplifying Expressions by Dividing

Simplifying expressions is an essential skill in calculus, especially when evaluating limits. One effective method is to divide each term by the highest degree term from the denominator. This trick helps to make expressions simpler, especially when dealing with infinite limits.

For the given exercise, we simplify the expression by dividing each term in the numerator and denominator by \(x^3\), the highest degree term. This yields: \[\frac{5 + \frac{1}{x^3}}{10 - \frac{3}{x} + \frac{7}{x^3}}\].

As \(x\) goes to infinity, terms like \(\frac{1}{x}, \frac{3}{x}\), and \(\frac{7}{x^3}\) approach zero, massively simplifying our equation to \(\frac{5}{10}\). This is a fundamental concept that allows reducing complex problems to simpler arithmetic ones.

Simplifying by dividing through helps not just in calculations, but it also provides clear insight into the behavior of functions as they are extended to their limits. This technique is a powerful tool in the calculus toolkit, providing both clarity and efficiency.