Chapter 2: Problem 6
Evaluate the following limits: \(\lim _{x \rightarrow \infty} \frac{x+1}{x^{3}-7 x+3}\)
Short Answer
Expert verified
Answer: The limit of the function is 0.
Step by step solution
01
Analyze the degrees of the numerator and the denominator
For the given function \(\frac{x+1}{x^3 - 7x + 3}\), we can see that the degree of the numerator is 1 and the degree of the denominator is 3.
02
Compare the degrees
Since the degree of the denominator is greater than the degree of the numerator (3 > 1), this tells us that when \(x\) approaches infinity, the denominator grows at a much faster rate than the numerator. Hence, the overall function will approach zero.
03
Determine the limit
Following the comparison in step 2, we find that the limit of the function as \(x\) approaches infinity is the limit as \(x\to\infty\) of \(\frac{x+1}{x^3 - 7x + 3}\):
$$\lim_{x \rightarrow \infty} \frac{x+1}{x^{3}-7 x+3} = 0$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Evaluating Limits
Understanding how to evaluate limits is a foundational skill in calculus. When evaluating the limit of a function as the variable approaches a certain value, you're essentially looking at what value the function is getting closer to as the input nears that point. In mathematical terms, when you see \( \lim_{x \rightarrow a} f(x) \) it's asking 'what does f(x) tend towards as x approaches a?'
To evaluate this, one common method is direct substitution, where you replace x with 'a' and simplify. However, when substitution leads to an undetermined form like 0/0 or \(/infty/\infty\), other strategies like factoring, using conjugates, expanding, or L'Hopital's Rule may be needed. In the case of our exercise, the direct comparison of polynomial degrees allows for a straightforward determination of the limit without complex manipulations.
To evaluate this, one common method is direct substitution, where you replace x with 'a' and simplify. However, when substitution leads to an undetermined form like 0/0 or \(/infty/\infty\), other strategies like factoring, using conjugates, expanding, or L'Hopital's Rule may be needed. In the case of our exercise, the direct comparison of polynomial degrees allows for a straightforward determination of the limit without complex manipulations.
Limits at Infinity
Limits at infinity involve finding the behavior of a function as the variable grows without bound, represented as \(x \rightarrow \pm\infty\). Such limits give us information about the end-behavior of functions.
For rational functions, like the one in our example (a fraction where both the numerator and the denominator are polynomials), analyzing the degrees of the polynomials can be particularly insightful. If the degree of the denominator is higher, the function approaches zero. Should the numerator have the higher degree, the function grows without bound, and if the degrees are equal, the function approaches the ratio of the leading coefficients. This understanding can be vital for graphing the long-term behavior of functions and accurately predicting their limits at infinity.
For rational functions, like the one in our example (a fraction where both the numerator and the denominator are polynomials), analyzing the degrees of the polynomials can be particularly insightful. If the degree of the denominator is higher, the function approaches zero. Should the numerator have the higher degree, the function grows without bound, and if the degrees are equal, the function approaches the ratio of the leading coefficients. This understanding can be vital for graphing the long-term behavior of functions and accurately predicting their limits at infinity.
Asymptotic Behavior
Asymptotic behavior describes how a function behaves as it gets close to a line or point, without actually reaching it. This concept is closely related to limits at infinity. As we have seen in the exercise, when a denominator grows faster than the numerator, the function's value gets closer and closer to zero — it's asymptotic to the x-axis.
An asymptote is a line that a graph approaches but doesn't touch. There are vertical asymptotes, often resulting from values that make the denominator of a fraction zero, and horizontal asymptotes, which are based on the end-behavior of the function as x approaches infinity. In the given problem, as \(x\) grows larger, the graph of the function \(\frac{x+1}{x^3 - 7x + 3}\) comes nearer to the x-axis, indicating a horizontal asymptote at y=0.
An asymptote is a line that a graph approaches but doesn't touch. There are vertical asymptotes, often resulting from values that make the denominator of a fraction zero, and horizontal asymptotes, which are based on the end-behavior of the function as x approaches infinity. In the given problem, as \(x\) grows larger, the graph of the function \(\frac{x+1}{x^3 - 7x + 3}\) comes nearer to the x-axis, indicating a horizontal asymptote at y=0.
Polynomial Functions
Polynomials are algebraic expressions composed of variables and coefficients, with the variables raised to only whole-number exponents. Each term in a polynomial has the form \(ax^n\) where 'a' is a non-zero coefficient, and 'n' is a non-negative integer as the exponent.
Polynomial functions exhibit nice continuous growth and do not have any sharp 'breaks' or 'holes' in their graphs. This makes evaluating their limits at infinity simpler, as seen in the example function. A significant aspect when working with polynomial functions is to look at the leading term, which is the term with the highest exponent, as it will dominate the behavior of the function when x becomes very large (positive or negative). The fundamental theorem of algebra tells us that a polynomial function of degree 'n' will have 'n' roots or solutions, which also translates into 'n - 1' turning points on its graph.
Polynomial functions exhibit nice continuous growth and do not have any sharp 'breaks' or 'holes' in their graphs. This makes evaluating their limits at infinity simpler, as seen in the example function. A significant aspect when working with polynomial functions is to look at the leading term, which is the term with the highest exponent, as it will dominate the behavior of the function when x becomes very large (positive or negative). The fundamental theorem of algebra tells us that a polynomial function of degree 'n' will have 'n' roots or solutions, which also translates into 'n - 1' turning points on its graph.
