Question

Find each limit, if possible. (a) \(\lim _{x \rightarrow \infty} \frac{x^{2}+2}{x^{3}-1}\) (b) \(\lim _{x \rightarrow \infty} \frac{x^{2}+2}{x^{2}-1}\) (c) \(\lim _{x \rightarrow \infty} \frac{x^{2}+2}{x-1}\)

Short answer

(a) The limit is 0. (b) The limit is 1. (c) The limit is infinity.

Step-by-step solution

Evaluate limit (a)

To find the limit of the function \(\lim _{x \rightarrow \infty} \frac{x^{2}+2}{x^{3}-1}\) as x tends to infinity, divide each term in the numerator and the denominator by \(x^{3}\) (the highest power of x in the denominator) which results in \(\lim _{x \rightarrow \infty} \frac{\frac{x^{2}}{x^{3}} + \frac{2}{x^{3}}}{1 - \frac{1}{x^{3}}}\). Simplify this to obtain \(\lim _{x \rightarrow \infty} \frac{\frac{1}{x} + \frac{2}{x^{3}}}{1 - \frac{1}{x^{3}}}\). As x approaches infinity, \(\frac{1}{x}\) and \(\frac{2}{x^{3}}\) approach 0, while \(\frac{1}{x^{3}}\) also approaches 0. Therefore, this limit is 0.

Evaluate limit (b)

Similarly for limit \(\lim _{x \rightarrow \infty} \frac{x^{2}+2}{x^{2}-1}\), divide each term by \(x^{2}\) (the highest power of x in the denominator). This results in \(\lim _{x \rightarrow \infty} \frac{1+\frac{2}{x^{2}}}{1-\frac{1}{x^{2}}}\). As x approaches infinity, both \(\frac{2}{x^{2}}\) and \(\frac{1}{x^{2}}\) approach 0. Therefore, the limit simplifies to \(\frac{1+0}{1-0}\), which equals 1.

Evaluate limit (c)

Finally, for the limit \(\lim _{x \rightarrow \infty} \frac{x^{2}+2}{x-1}\), the highest power of x is found in the numerator. When the degree of the polynomial in the numerator is higher than that in the denominator and the limit is approached at infinity, the limit is either positive or negative infinity depending on the sign of the coefficients of the highest degree term. In this case, as x approaches infinity, the value of the function becomes larger and larger without bound. Therefore, this limit is positive infinity.

Summarize the results

To summarize, the solutions to the limits are: (a) 0, (b) 1 and (c) infinity. When the degree of the polynomial function in the denominator is higher, the limit is 0; when degrees are equal, the limit is the coefficient of the highest degree terms; and when the degree is higher in the numerator, the limit is infinity.

Key concepts

Infinity

Infinity is a concept that represents an unbounded quantity that's larger than any real number. In mathematics, infinity is often denoted by the symbol \( \infty \). It is not a number in the traditional sense but rather an idea that helps us understand and describe behavior that goes beyond finite limits. When we talk about limits as \( x \rightarrow \infty \), we're examining what happens to a function as the value of \( x \) increases without bound. In other words, we want to see the trend of the function as \( x \) becomes incredibly large.

• If the function heads towards a particular value, we attribute that value to the limit.
• If it grows significantly large, the limit may be infinity.

The same principle applies when \( x \rightarrow -\infty \), but in that case, we're observing the behavior as \( x \) heads towards negative infinity.

Polynomials

Polynomials are expressions that consist of variables, coefficients, and the operations of addition, subtraction, multiplication, and non-negative integer exponents. The general form of a polynomial is: \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \) Where:

• \( a_n, a_{n-1}, \ldots, a_0 \) are coefficients.
• \( n \) is the degree of the polynomial, which we'll discuss further in another section.

Polynomials are pivotal in calculus and algebra because they are easy to manipulate and provide straightforward models for comparison or approximation of more complex mathematical functions.When working with limits involving polynomials, it's useful to compare the highest power of \( x \) in both the numerator and the denominator. This helps simplify the problem by leveraging the dominant behavior of the polynomial.

Degree of Polynomials

The degree of a polynomial is determined by the highest power of the variable \( x \) with a non-zero coefficient in that polynomial. In the polynomial \( a_n x^n + \ldots + a_0 \), the degree is \( n \). It provides valuable information about the function's behavior, especially when evaluating limits as \( x \rightarrow \infty \).

• If the degree of the polynomial in the numerator is higher than in the denominator, the limit tends towards infinity.
• If both polynomials have the same degree, the limit is the ratio of the leading coefficients.
• If the degree is higher in the denominator, the limit tends towards zero.

This understanding enables us to predict the kind of infinity (positive or negative) or the finite value towards which a given polynomial expression will tend as \( x \) becomes infinitely large.