Problem 20
Question
Use the Infinite Limit Theorem and the properties of limits as in Example 6 to find the horizontal asymptotes (if any) of the graph of the given function. $$g(x)=\frac{2 x^{5}-x^{3}+2 x-9}{5-x^{5}}$$
Step-by-Step Solution
Verified Answer
Answer: The horizontal asymptote of the given function is \(y = -2\).
1Step 1: Rewrite the function
Rewrite the given function as:
$$g(x) = \frac{2 x^5 - x^3 + 2x - 9}{5 - x^5}$$
2Step 2: Calculate the limit as x approaches infinity
Calculate the limit of the function as x approaches infinity by dividing both the numerator and the denominator by the highest power of x (in this case, \(x^5\)):
$$\lim_{x \rightarrow \infty} \frac{2\frac{x^5}{x^5}- \frac{x^3}{x^5} + \frac {2x}{x^5} - \frac{9}{x^5}}{\frac{5}{x^5}-\frac{x^5}{x^5}}$$
Now simplify the expression:
$$\lim_{x \rightarrow \infty} \frac{2-\frac{1}{x^2}+\frac{2}{x^4}-\frac{9}{x^5}}{\frac{5}{x^5}-1}$$
As \(x\) approaches infinity, all terms with \(x\) in the denominator will approach zero. Thus, the limit becomes:
$$\lim_{x \rightarrow \infty} \frac{2 - 0 + 0 - 0}{0 - 1} = \frac{2}{-1} = -2$$
3Step 3: Calculate the limit as x approaches negative infinity
Calculate the limit of the function as x approaches negative infinity just like in Step 2:
$$\lim_{x \rightarrow -\infty} \frac{2\frac{x^5}{x^5}-\frac{x^3}{x^5}+\frac{2x}{x^5}-\frac{9}{x^5}}{\frac{5}{x^5}-\frac{x^5}{x^5}}$$
Now simplify the expression:
$$\lim_{x \rightarrow -\infty} \frac{2-\frac{1}{x^2}+\frac{2}{x^4}-\frac{9}{x^5}}{\frac{5}{x^5}-1}$$
As \(x\) approaches negative infinity, all terms with \(x\) in the denominator will approach zero. Thus, the limit becomes:
$$\lim_{x \rightarrow -\infty} \frac{2 - 0 + 0 - 0}{0 - 1} = \frac{2}{-1} = -2$$
4Step 4: Determine the horizontal asymptotes
Since the limit of the function as \(x\) approaches both infinity and negative infinity equals -2, there is a horizontal asymptote at \(y = -2\).
The horizontal asymptote(s) of the given function is:
$$y = -2$$
Key Concepts
Infinite Limit TheoremProperties of LimitsAsymptotic BehaviorLimit of a Function
Infinite Limit Theorem
Understanding the behavior of a function as the input value grows without bound is crucial in calculus and is where the Infinite Limit Theorem comes into play.
The Infinite Limit Theorem helps us determine how a function behaves as the independent variable approaches infinity or negative infinity. It states that if the degrees of the polynomial in the numerator and denominator are equal, the horizontal asymptote of the function can be found by dividing the leading coefficients. In the given example, we deal with the function
\[g(x)=\frac{2 x^{5}-x^{3}+2 x-9}{5-x^{5}}\]
where both the numerator and the denominator are of degree five. Thus, we apply the theorem to find the horizontal asymptotes by analyzing the limits as x tends towards infinity and negative infinity.
The Infinite Limit Theorem helps us determine how a function behaves as the independent variable approaches infinity or negative infinity. It states that if the degrees of the polynomial in the numerator and denominator are equal, the horizontal asymptote of the function can be found by dividing the leading coefficients. In the given example, we deal with the function
\[g(x)=\frac{2 x^{5}-x^{3}+2 x-9}{5-x^{5}}\]
where both the numerator and the denominator are of degree five. Thus, we apply the theorem to find the horizontal asymptotes by analyzing the limits as x tends towards infinity and negative infinity.
Properties of Limits
When we approach the evaluation of limits, the properties of limits provide us with a reliable toolkit. These properties include operations that allow us to add, subtract, multiply, and divide limits, as well as power and root laws. However, a significant caution is necessary: these operations can only be used when the limits exist.
In our function example, we utilize these properties after dividing the terms by the highest power of x in the function. By doing so, we simplify the function and examine the behavior of the remaining terms as x approaches infinity. This approach adheres to the property of limits that allows us to divide by powers of x and see that terms become negligible as x approaches infinity.
In our function example, we utilize these properties after dividing the terms by the highest power of x in the function. By doing so, we simplify the function and examine the behavior of the remaining terms as x approaches infinity. This approach adheres to the property of limits that allows us to divide by powers of x and see that terms become negligible as x approaches infinity.
Asymptotic Behavior
The term asymptotic behavior refers to the behavior of a function as it gets closer to a line called an asymptote without ever quite reaching it. In particular, horizontal asymptotes describe the behavior of functions as x approaches infinity or negative infinity.
In the given exercise, we seek horizontal asymptotes to understand where the function g(x) 'levels off' at extreme values of x. The calculations we performed inform us that, regardless of whether x goes to positive or negative infinity, the function approaches the constant value y = -2. This constant becomes the horizontal asymptote, conveying that as x grows larger in magnitude, the function will come closer and closer to y = -2 but will never actually be equal to -2.
In the given exercise, we seek horizontal asymptotes to understand where the function g(x) 'levels off' at extreme values of x. The calculations we performed inform us that, regardless of whether x goes to positive or negative infinity, the function approaches the constant value y = -2. This constant becomes the horizontal asymptote, conveying that as x grows larger in magnitude, the function will come closer and closer to y = -2 but will never actually be equal to -2.
Limit of a Function
The limit of a function is a fundamental concept in calculus that tells us the value that a function approaches as the input (or x-value) approaches a certain point. It's not about the value of the function at that point, but about where it's heading as it gets arbitrarily close to that point.
Through the exercise provided, we've seen the practical application of finding a limit when x approaches infinity. It's crucial to recognize that, while sometimes a function's limit can be calculated by merely substituting the value of x, in cases involving infinity, we need to apply theorems and properties of limits. For example, when x approaches infinity in our function g(x), the limit is -2. Knowing this behavior gives us an insight into the function's growth and can vastly influence how we interpret or utilize the function in various applied mathematics fields.
Through the exercise provided, we've seen the practical application of finding a limit when x approaches infinity. It's crucial to recognize that, while sometimes a function's limit can be calculated by merely substituting the value of x, in cases involving infinity, we need to apply theorems and properties of limits. For example, when x approaches infinity in our function g(x), the limit is -2. Knowing this behavior gives us an insight into the function's growth and can vastly influence how we interpret or utilize the function in various applied mathematics fields.
Other exercises in this chapter
Problem 19
Determine whether or not the function is continuous at the given number. $$f(x)=\left\\{\begin{array}{cl} -2 x+4 & \text { if } x \leq 2 \\ 2 x-4 & \text { if }
View solution Problem 19
Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 3^{+}} \frac{3}{x^{2}-9}$$
View solution Problem 20
Determine whether or not the function is continuous at the given number. $$g(x)=\left\\{\begin{array}{cl} 2 x+5 & \text { if } x
View solution Problem 20
Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 2^{-}} \frac{x+1}{x^{2}-x-2}$$
View solution