Problem 26
Question
Determine the following limits. $$\lim _{x \rightarrow \infty} \frac{9 x^{3}+x^{2}-5}{3 x^{4}+4 x^{2}}$$
Step-by-Step Solution
Verified Answer
Answer: The limit of the expression as x approaches infinity is 0.
1Step 1: Identify the dominant term in the numerator and the denominator
In the given expression:
$$\frac{9 x^{3}+x^{2}-5}{3 x^{4}+4 x^{2}}$$
The dominant term in the numerator is \(9x^3\) and that in the denominator is \(3x^4\).
2Step 2: Divide all terms by the highest power of x (#latex# x^4#/#latex#)
Divide all terms in the numerator and the denominator by \(x^4\). This will simplify the expression and help in evaluating the limit. The expression then becomes:
$$\frac{\frac{9 x^{3}}{x^{4}}+\frac{x^{2}}{x^{4}}-\frac{5}{x^{4}}}{\frac{3 x^{4}}{x^{4}}+\frac{4 x^{2}}{x^{4}}}$$
3Step 3: Simplify the expression
Simplify the expression by canceling out the common factors in the fractions:
$$\frac{9\left(\frac{x^{3}}{x^{4}}\right)+\frac{x^{2}}{x^{4}}-\frac{5}{x^{4}}}{3\left(\frac{x^{4}}{x^{4}}\right)+4\left(\frac{x^{2}}{x^{4}}\right)}=\frac{9\left(\frac{1}{x}\right)+\frac{1}{x^{2}}-\frac{5}{x^{4}}}{3+4\left(\frac{1}{x^{2}}\right)}$$
4Step 4: Evaluate the limit as x approaches infinity
As x tends to infinity, all the fractions with x in the denominator will approach 0.
$$\lim _{x \rightarrow \infty} \frac{9\left(\frac{1}{x}\right)+\frac{1}{x^{2}}-\frac{5}{x^{4}}}{3+4\left(\frac{1}{x^{2}}\right)} = \lim _{x \rightarrow \infty} \frac{9(0)+0-0}{3+4(0)} = \frac{0}{3} = 0$$
So, the limit of the given expression as x approaches infinity is 0.
Key Concepts
Limits at InfinityDominant Term in a PolynomialSimplifying Expressions
Limits at Infinity
When studying calculus, understanding limits at infinity is crucial for grasping the behavior of functions as the input grows without bound. To evaluate a limit at infinity, the task is to determine what value a function approaches as the variable tends toward either positive or negative infinity.
Limits at infinity often involve fractions where the numerator and denominator are polynomials, as in the textbook exercise. In these cases, if the degrees of the numerator and the denominator polynomials are different, as the value of x gets larger, the terms with the highest powers of x become the most significant. In particular, if the degree of the numerator is less than the degree of the denominator, the limit will be zero, which is what we see in the provided example. However, if the degree is greater in the numerator, the limit tends towards finity (or -finity, depending on the leading coefficients), and if the degrees are equal, the limit will be the ratio of the leading coefficients.
In practice, to evaluate these limits, we often simplify the expression so that the dominant terms' behavior dictates the limit value as x approaches infinity.
Limits at infinity often involve fractions where the numerator and denominator are polynomials, as in the textbook exercise. In these cases, if the degrees of the numerator and the denominator polynomials are different, as the value of x gets larger, the terms with the highest powers of x become the most significant. In particular, if the degree of the numerator is less than the degree of the denominator, the limit will be zero, which is what we see in the provided example. However, if the degree is greater in the numerator, the limit tends towards finity (or -finity, depending on the leading coefficients), and if the degrees are equal, the limit will be the ratio of the leading coefficients.
In practice, to evaluate these limits, we often simplify the expression so that the dominant terms' behavior dictates the limit value as x approaches infinity.
Dominant Term in a Polynomial
Within any polynomial, the term with the highest power of the variable is known as the dominant term. This term becomes increasingly significant as the variable approaches infinity or negative infinity.
For example, in the polynomial 9x^3 + x^2 - 5, the term 9x^3 is the dominant term because it has the highest power of x. The dominance of this term becomes apparent as x grows very large or very small; the value of the polynomial will be largely determined by the value of 9x^3.
This concept is especially important when evaluating limits at infinity. When we have a rational function—where both the numerator and the denominator are polynomials—the rate of growth of the polynomials as x becomes large determines the limit of the function. Hence, identifying the dominant term in both the numerator and the denominator is a key step in finding limits at infinity.
For example, in the polynomial 9x^3 + x^2 - 5, the term 9x^3 is the dominant term because it has the highest power of x. The dominance of this term becomes apparent as x grows very large or very small; the value of the polynomial will be largely determined by the value of 9x^3.
This concept is especially important when evaluating limits at infinity. When we have a rational function—where both the numerator and the denominator are polynomials—the rate of growth of the polynomials as x becomes large determines the limit of the function. Hence, identifying the dominant term in both the numerator and the denominator is a key step in finding limits at infinity.
Simplifying Expressions
To simplify expressions, particularly when dealing with limits at infinity, we aim to reduce the complexity of the expression while maintaining its mathematical integrity. Simplification often involves factoring, expanding, canceling common factors, and dividing each term by the highest power of x found in the denominator when dealing with fractions.
In our example, dividing by the power of x found in the highest term of the denominator x^4 makes it easier to see what happens as x approaches infinity. After simplification, any terms with x in the denominator will approach zero, making it evident what the value of the function will be at infinity.
Ultimately, simplifying expressions aids in understanding and calculating limits. It is often a necessary step to reveal the true behavior of a function as variables approach certain critical values, such as infinity, thus providing a clearer path towards the solution.
In our example, dividing by the power of x found in the highest term of the denominator x^4 makes it easier to see what happens as x approaches infinity. After simplification, any terms with x in the denominator will approach zero, making it evident what the value of the function will be at infinity.
Ultimately, simplifying expressions aids in understanding and calculating limits. It is often a necessary step to reveal the true behavior of a function as variables approach certain critical values, such as infinity, thus providing a clearer path towards the solution.
Other exercises in this chapter
Problem 26
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