Chapter 23: Problem 19
Limits Involving Zero or Infinity $$\lim _{x \rightarrow \infty} \frac{2 x+5}{x-4}$$
Short Answer
Expert verified
The limit as x approaches infinity of \(\frac{2x+5}{x-4}\) is 2.
Step by step solution
01
Identify the Dominant Terms
Recognize that as x approaches infinity, the terms with the highest powers in the numerator and denominator dominate the behavior of the fraction. In this case, the dominant term in the numerator is 2x and the dominant term in the denominator is x.
02
Divide by the Dominant Terms
Divide the numerator and denominator by the dominant term of the denominator (x). The expression becomes: \(\lim _{x \rightarrow \infty} \frac{2x/x+5/x}{x/x-4/x}\).
03
Simplify the Expression
Simplify the fraction by canceling the x terms and evaluating the resulting limits as x approaches infinity. The expression simplifies to: \(\lim _{x \rightarrow \infty} \frac{2+5/x}{1-4/x}\).
04
Evaluate the Limits
As x approaches infinity, the terms containing 1/x approach 0. The expression simplifies to: \(\lim _{x \rightarrow \infty} \frac{2+0}{1-0}\).
05
Find the Limit
The final simplified expression is \(\frac{2}{1}\). Therefore, the limit is 2.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limit of a Function
Understanding the limit of a function is a fundamental concept in calculus, particularly when dealing with functions that exhibit extreme behavior, such as growth towards infinity. It is a way of describing the behavior of a function as the input (or 'x' value) approaches a certain point, which can be a finite number, zero, or infinity.
In the provided exercise, the task is to find the limit as 'x' approaches infinity for the function \(\frac{2x+5}{x-4}\). To tackle this, one must recognize that as 'x' becomes very large, certain terms in the function become much more significant than others. This observation leads to the simplification of the function by removing less significant terms, allowing us to focus on the dominant parts of the function as 'x' approaches infinity.
The final answer reveals the horizontal asymptote of the function, a concept intertwined with the function's limit at infinity. By analyzing the behavior of the dominant terms, we can deduce that the function 'levels out' at a certain value, in this case, 2, when 'x' is very large.
In the provided exercise, the task is to find the limit as 'x' approaches infinity for the function \(\frac{2x+5}{x-4}\). To tackle this, one must recognize that as 'x' becomes very large, certain terms in the function become much more significant than others. This observation leads to the simplification of the function by removing less significant terms, allowing us to focus on the dominant parts of the function as 'x' approaches infinity.
The final answer reveals the horizontal asymptote of the function, a concept intertwined with the function's limit at infinity. By analyzing the behavior of the dominant terms, we can deduce that the function 'levels out' at a certain value, in this case, 2, when 'x' is very large.
Asymptotic Behavior
The asymptotic behavior of a function pertains to its end behavior as the input value grows larger and larger or smaller and smaller (approaching infinity or negative infinity, respectively). An asymptote is a line that the graph of the function approaches but never quite reaches. The classic cases include horizontal, vertical, and slant (oblique) asymptotes.
In regards to horizontal asymptotes, these occur when the 'y' value of a function approaches a constant as 'x' approaches infinity or negative infinity. For instance, in our exercise, the horizontal asymptote is determined by the limit we calculated, which is 2. This means as 'x' grows without bound, the graph of our function will get closer and closer to the line 'y=2' but will never actually touch it. The understanding of a function’s asymptotic behavior is crucial for graphing functions and understanding their long-term behavior.
In regards to horizontal asymptotes, these occur when the 'y' value of a function approaches a constant as 'x' approaches infinity or negative infinity. For instance, in our exercise, the horizontal asymptote is determined by the limit we calculated, which is 2. This means as 'x' grows without bound, the graph of our function will get closer and closer to the line 'y=2' but will never actually touch it. The understanding of a function’s asymptotic behavior is crucial for graphing functions and understanding their long-term behavior.
Rational Functions
A rational function is a ratio of two polynomials. It looks like \(\frac{P(x)}{Q(x)}\), where both P and Q are polynomials and Q is not the zero polynomial. These functions can exhibit a variety of behaviors, such as vertical asymptotes, horizontal asymptotes, and holes depending on the relationship between P(x) and Q(x).
The exercise presented involves a rational function where the degree of the numerator and denominator are the same. When this happens, and 'x' goes towards infinity, the function will approach the ratio of the leading coefficients, which is visible through the process of dividing by the highest power of 'x' found in the denominator.
It is important to note that simplification through division by 'x' is a common method to find horizontal asymptotes for rational functions with equal degrees in the numerator and denominator. The zeroes and undefined points of rational functions can often be found by setting the numerator equal to zero and the denominator equal to zero, respectively, although doing so for finding asymptotic behavior at infinity is usually unnecessary.
The exercise presented involves a rational function where the degree of the numerator and denominator are the same. When this happens, and 'x' goes towards infinity, the function will approach the ratio of the leading coefficients, which is visible through the process of dividing by the highest power of 'x' found in the denominator.
It is important to note that simplification through division by 'x' is a common method to find horizontal asymptotes for rational functions with equal degrees in the numerator and denominator. The zeroes and undefined points of rational functions can often be found by setting the numerator equal to zero and the denominator equal to zero, respectively, although doing so for finding asymptotic behavior at infinity is usually unnecessary.
