Find the Asymptotes f(x)=x/( fourth root of x^4+1)

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 $\underset{x\to \mathrm{\infty }}{lim}\frac{x}{\sqrt[4]{x^4+1}}$.

Step 3.1:

Normalize the fraction by dividing both the numerator and denominator by $x$, the highest power of $x$ in the denominator: $\underset{x\to \mathrm{\infty }}{lim}\frac{\frac{x}{x}}{\sqrt[4]{\frac{x^4}{x^4}+\frac{1}{x^4}}}$.

Step 3.2:

Proceed to evaluate the limit.

Step 3.2.1:

Simplify by canceling the $x$ in the numerator: $\underset{x\to \mathrm{\infty }}{lim}\frac{1}{\sqrt[4]{\frac{x^4}{x^4}+\frac{1}{x^4}}}$.

Step 3.2.2:

Further simplify by canceling the $x^4$ in the denominator.

Step 3.2.2.1:

Remove the common factors: $\underset{x\to \mathrm{\infty }}{lim}\frac{1}{\sqrt[4]{1+\frac{1}{x^4}}}$.

Step 3.2.2.2:

Rewrite the expression for clarity: $\underset{x\to \mathrm{\infty }}{lim}\frac{1}{\sqrt[4]{1+\frac{1}{x^4}}}$.

Step 3.2.3:

Apply the Limits Quotient Rule: $\frac{\underset{x\to \mathrm{\infty }}{lim}1}{\underset{x\to \mathrm{\infty }}{lim}\sqrt[4]{1+\frac{1}{x^4}}}$.

Step 3.2.4:

Evaluate the limit of the constant 1: $\frac{1}{\underset{x\to \mathrm{\infty }}{lim}\sqrt[4]{1+\frac{1}{x^4}}}$.

Step 3.2.5:

Move the limit inside the radical: $\frac{1}{\sqrt[4]{\underset{x\to \mathrm{\infty }}{lim}1+\frac{1}{x^4}}}$.

Step 3.2.6:

Apply the Sum of Limits Rule: $\frac{1}{\sqrt[4]{\underset{x\to \mathrm{\infty }}{lim}1+\underset{x\to \mathrm{\infty }}{lim}\frac{1}{x^4}}}$.

Step 3.2.7:

Evaluate the limit of the constant 1: $\frac{1}{\sqrt[4]{1+\underset{x\to \mathrm{\infty }}{lim}\frac{1}{x^4}}}$.

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:

1. Domain of a Function: The set of all possible input values (x-values) for which the function is defined.

2. 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.

3. 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.

4. 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.

5. 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.

6. Sum of Limits Rule: The limit of a sum is equal to the sum of the limits, provided the limits exist.

7. 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.