Question

In Exercises \(11-14,\) find each limit, if possible. (a) \(\lim _{x \rightarrow \infty} \frac{5-2 x^{3 / 2}}{3 x^{2}-4}\) (b) \(\lim _{x \rightarrow \infty} \frac{5-2 x^{3 / 2}}{3 x^{3 / 2}-4}\) (c) \(\lim _{x \rightarrow \infty} \frac{5-2 x^{3 / 2}}{3 x-4}\)

Short answer

The limits of the functions a), b) and c) are 0, -2/3 and 0 respectively.

Step-by-step solution

Finding limit for (a)

This is approached by expressing the numerator and denominator by their largest exponents. Rewrite the fraction as \(\lim _{x \rightarrow \infty} \frac{2 x^{3 / 2} ( \frac{5}{2 x^{3 / 2}} - 1 )}{3 x^{2} ( 1 - \frac{4}{3x^{2}} )}\). On further simplification, the limit becomes \(\lim _{x \rightarrow \infty} \frac{2 x^{ -1/2} ( \frac{5}{2 x^{3 / 2}} - 1 )}{3 ( 1 - \frac{4}{3x^{2}} )} \) which tends to 0 as x tends to infinity.

Finding limit for (b)

Here the numerator and denominator have the same exponent, so we can evaluate the limit by evaluating the coefficient ratio which gives \(\frac{-2}{3}\). This is because when the highest powers of x in the numerator and denominator are the same, the limit is the ratio of the coefficients of these terms.

Finding limit for (c)

Please keep in mind that when the highest power of x in the numerator is higher than that in the denominator, the limit is +/- infinity depending on the coefficients of these terms. In the case where the power of x in the numerator is lower than in the denominator, the limit is 0. We apply this to find that the limit is 0 as the power of x in the denominator is higher than in the numerator.

Key concepts

Calculus

Calculus is an area of mathematics that studies continuous change and includes the study of limits, derivatives, integrals, and infinite series. The concept of a limit is fundamental in calculus and gives rise to the derivative and the integral, which are the two main operations in calculus. It explores the behavior of functions as they approach a specific point, either numerically or symbolically. For instance, the limits in the given exercise help us understand how the function behaves as the variable approaches infinity, providing insight into the growth or decay of the function components.

Calculus is widely used in science, engineering, economics, and other fields, offering a framework for modeling and solving problems involving dynamic systems. In our everyday exercise, we apply calculus concepts to evaluate the behavior of rational functions as an input variable grows without bound, showcasing calculus' power in analyzing trends and predicting outcomes.

Infinite Limits

Infinite limits occur when the value of a function increases or decreases without bound as the variable approaches a certain value, or infinity. This concept is crucial in understanding the long-term behavior of functions. For example, when the variable approaches infinity, as in the exercise, we are observing the behavior of the function at very large values of the variable.

There are special considerations when dealing with infinite limits. If the degrees of the terms in the numerator and denominator of a rational function are different, we can often determine the limit by looking at the terms with the highest powers. In our case, exercise (a) tends to 0, as the denominator increases much faster than the numerator, meaning the fractions value becomes negligible as we choose larger and larger values of x. Understanding infinite limits allows us to analyze and predict behavior in fields varying from physics to finance, where variables can stretch towards incredibly large or incredibly small values.

Limit Evaluation Methods

There are several methods for evaluating limits, and identifying the best approach often depends on the function's form. For polynomial and rational functions, one might simply observe the coefficients and powers of the highest degree terms, as illustrated in the textbook solution.

In methodical terms, if both the numerator and the denominator tend to infinity, we commonly divide each term by the highest power of x that appears in the denominator to simplify the expression. This technique, called 'division by highest power', was used in step 1 of the solution to approach limit (a). For exercise (b), because the terms were already of the same degree, the limit was the ratio of their coefficients, demonstrating the 'leading coefficient rule'. These primary techniques, along with others, form a toolbox for students tackling diverse limits in calculus problems. Understanding when and how to apply these strategies can save time and reduce the complexity involved in finding limits of functions.