Find the Asymptotes f(x)=x/( fourth root of x^4+1)
The problem is asking for the determination of the asymptotes of the function f(x) = x / (⁴√(x⁴ + 1)). An asymptote is a line that the graph of the function approaches but never actually reaches. There are different types of asymptotes, namely vertical, horizontal, and oblique (slant).
Vertical asymptotes occur where the function goes to infinity as x approaches a particular value. Horizontal asymptotes represent the value that f(x) approaches as x goes to positive or negative infinity. Oblique asymptotes are diagonal lines that the function approaches as x goes to infinity or negative infinity.
The problem requires identifying any such asymptotes by examining the behavior of the function as x approaches various critical points, such as x tending to infinity, negative infinity, or points where the denominator of the function might be zero (which isn’t the case in this particular function since the denominator x⁴ + 1 is always positive).
$f \left(\right. x \left.\right) = \frac{x}{\sqrt[4]{x^{4} + 1}}$
Answer
Solution:
Step 1:
Determine the domain of $\frac{x}{\sqrt[4]{x^{4} + 1}}$ by identifying values of $x$ that cause the denominator to be undefined. There are no such values, so the domain is all real numbers.
Step 2:
Identify any vertical asymptotes by locating points of infinite discontinuity. There are no vertical asymptotes for this function.
Step 3:
To find the horizontal asymptote, calculate $\lim_{x \to \infty} \frac{x}{\sqrt[4]{x^{4} + 1}}$.
Step 3.1:
Normalize the fraction by dividing both the numerator and denominator by $x$, the highest power of $x$ in the denominator: $\lim_{x \to \infty} \frac{\frac{x}{x}}{\sqrt[4]{\frac{x^{4}}{x^{4}} + \frac{1}{x^{4}}}}$.
Step 3.2:
Proceed to evaluate the limit.
Step 3.2.1:
Simplify by canceling the $x$ in the numerator: $\lim_{x \to \infty} \frac{1}{\sqrt[4]{\frac{x^{4}}{x^{4}} + \frac{1}{x^{4}}}}$.
Step 3.2.2:
Further simplify by canceling the $x^{4}$ in the denominator.
Step 3.2.2.1:
Remove the common factors: $\lim_{x \to \infty} \frac{1}{\sqrt[4]{1 + \frac{1}{x^{4}}}}$.
Step 3.2.2.2:
Rewrite the expression for clarity: $\lim_{x \to \infty} \frac{1}{\sqrt[4]{1 + \frac{1}{x^{4}}}}$.
Step 3.2.3:
Apply the Limits Quotient Rule: $\frac{\lim_{x \to \infty} 1}{\lim_{x \to \infty} \sqrt[4]{1 + \frac{1}{x^{4}}}}$.
Step 3.2.4:
Evaluate the limit of the constant 1: $\frac{1}{\lim_{x \to \infty} \sqrt[4]{1 + \frac{1}{x^{4}}}}$.
Step 3.2.5:
Move the limit inside the radical: $\frac{1}{\sqrt[4]{\lim_{x \to \infty} 1 + \frac{1}{x^{4}}}}$.
Step 3.2.6:
Apply the Sum of Limits Rule: $\frac{1}{\sqrt[4]{\lim_{x \to \infty} 1 + \lim_{x \to \infty} \frac{1}{x^{4}}}}$.
Step 3.2.7:
Evaluate the limit of the constant 1: $\frac{1}{\sqrt[4]{1 + \lim_{x \to \infty} \frac{1}{x^{4}}}}$.
Step 3.3:
Since the fraction $\frac{1}{x^{4}}$ has an unbounded denominator, it approaches 0: $\frac{1}{\sqrt[4]{1 + 0}}$.
Step 3.4:
Simplify the expression.
Step 3.4.1:
Simplify the denominator.
Step 3.4.1.1:
Combine 1 and 0: $\frac{1}{\sqrt[4]{1}}$.
Step 3.4.1.2:
The fourth root of 1 is 1: $\frac{1}{1}$.
Step 3.4.2:
Divide 1 by 1 to get the final result: $1$.
Step 4:
The horizontal asymptote is $y = 1$.
Step 5:
Attempt to find any oblique asymptotes using polynomial division. However, due to the presence of a radical, polynomial division is not applicable. No oblique asymptotes can be found.
Step 6:
Compile the set of all asymptotes for the function:
- No Vertical Asymptotes
- Horizontal Asymptotes: $y = 1$
- No Oblique Asymptotes
Step 7:
End of the problem-solving process.
Knowledge Notes:
Domain of a Function: The set of all possible input values (x-values) for which the function is defined.
Vertical Asymptotes: These occur at values of x where the function approaches infinity. They are found by setting the denominator equal to zero and solving for x.
Horizontal Asymptotes: These are horizontal lines that the graph of a function approaches as x goes to infinity or negative infinity. They are found by evaluating the limit of the function as x approaches infinity or negative infinity.
Limits and Asymptotes: Limits are used to determine the behavior of a function as it approaches a certain value or infinity. They are essential in finding horizontal asymptotes.
Limits Quotient Rule: If the limits of the numerator and denominator exist, the limit of the quotient is the quotient of the limits, provided the limit of the denominator is not zero.
Sum of Limits Rule: The limit of a sum is equal to the sum of the limits, provided the limits exist.
Polynomial Division: A method used to simplify functions and find oblique asymptotes. It is not applicable when the function includes radicals or other non-polynomial expressions.
