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Chapter 3: Problem 32

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

Short Answer

Expert verified
The value of the limit as x tends to infinity is 0.5.

Step by step solution

01

Identify Highest Power of x

Identify the highest power of x in both numerator and denominator. Here, that's \(x^3\). By rule of limit for rational function, the limit as x tends to infinity or minus infinity is the ratio of the coefficients of the terms with the highest power of x.
02

Calculate the ratio of coefficients

The coefficient of \(x^{3}\) in the numerator is 5, and in the denominator is 10. Therefore, the limit is the ratio of these coefficients which equals to \( \frac{5}{10}\).
03

Simplify the ratio

Simplify the ratio to get the value of the limit. Here, \( \frac{5}{10}\) simplifies to 0.5 .

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Rational Functions
When we talk about rational functions in calculus, we refer to any function that can be expressed as the ratio of two polynomials. For instance, the function given in the exercise \( f(x) = \frac{5x^3 + 1}{10x^3 - 3x^2 + 7} \) is a rational function because it is composed of a polynomial in the numerator and another in the denominator.
Asymptotic Behavior of Rational Functions
Asymptotic behavior describes how a function behaves as it approaches a particular point or infinity. In this case, we are interested in the behavior of our rational function as \( x \) approaches infinity. Our goal is to identify if the function approaches a specific value (the limit), becomes unbounded, or oscillates without approaching any particular value. This concept is crucial as it gives us insights into the end behavior of functions, which is essential in many branches of mathematics and applied sciences.
Coefficient Ratio and Limits
The coefficient ratio is particularly useful when dealing with limits of rational functions as \( x \) approaches infinity or minus infinity. If the degrees of the highest power terms in the numerator and denominator are the same, the limit will be the ratio of the coefficients of those terms. Hence, in the given exercise, the limit of the function as \( x \) approaches infinity is determined by dividing the coefficient of \( x^3 \) in the numerator (5) by the coefficient of \( x^3 \) in the denominator (10), resulting in a limit of 0.5.

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Most popular questions from this chapter

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