Question

Determine $limf(x)$ and $\underset{x\to \mathrm{\infty }}{lim}f(x)$ for the following functions. Then give the horizontal asymptote$(\text{s) of }f$ (if any). $$f(x)=\frac{4x^3+1}{2x^3+\sqrt{16x^6+1}}$$

Short answer

Answer: The limit of the function as $x$ approaches infinity is 0, and the horizontal asymptote is $y=0$.

Step-by-step solution

Identify the highest power of x in the numerator and the denominator

In this case, the highest power is $x^3$ in the numerator and $x^6$ in the denominator. Write the function as: $$f(x)=\frac{4x^3+1}{2x^3+\sqrt{16x^6+1}}$$

Divide every term by the highest power in the denominator

In this case, we will divide each term by $x^6$. The new function will be: $$f(x)=\frac{\frac{4x^3}{x^6}+\frac{1}{x^6}}{\frac{2x^3}{x^6}+\sqrt{\frac{16x^6}{x^6}+\frac{1}{x^6}}}$$

Simplify the terms

We'll simplify the terms in the fraction by reducing the powers of x: $$f(x)=\frac{\frac{4}{x^3}+\frac{1}{x^6}}{\frac{2}{x^3}+\sqrt{16+\frac{1}{x^6}}}$$

Evaluate the limit as $x\to \mathrm{\infty }$

As $x$ approaches infinity, every term with a power of x in the denominator approaches zero. So, we have: $$\underset{x\to \mathrm{\infty }}{lim}f(x)=\frac{0+0}{0+\sqrt{16+0}}=\frac{0}{\sqrt{16}}$$ Therefore, $$\underset{x\to \mathrm{\infty }}{lim}f(x)=0$$

Find the horizontal asymptote

Since the limit as $x$ approaches infinity is 0, there is only one horizontal asymptote: $$y=0$$ In conclusion, the limit of function $f(x)$ as $x$ approaches infinity is 0, and the horizontal asymptote is $y=0$.

Key concepts

Horizontal Asymptote

In the realm of calculus, the horizontal asymptote of a function is a horizontal line that the function's graph approaches as the input (or 'x' value) heads towards infinity or negative infinity. These asymptotes represent the behavior of a function at the outermost edges of the x-axis.

Understanding horizontal asymptotes is crucial when dealing with limits at infinity. For the given function $f(x)=\frac{4x^3+1}{2x^3+\sqrt{16x^6+1}}$, after simplification, we notice that as $x$ increases without bound, the terms $\frac{1}{x^6}$ and $\frac{4}{x^3}$ approach zero. In essence, the function's value approaches the constant term in the denominator's square root, which in this case is 16. The square root of 16 is 4, and thus, as $x$ becomes very large, the function approaches $0/4$, which simplifies to $0$. This means that the horizontal asymptote is the line $y=0$ or the x-axis, a conclusion we reach by identifying the limit of the function as $x$ approaches infinity.

This understanding also helps predict and sketch the long-run behavior of the function's graph, an essential aspect of curve sketching in calculus.

Polynomial Functions

Polynomial functions are among the simplest and most commonly encountered types of functions in algebra and calculus. They are formed by the sum of terms consisting of a variable raised to a non-negative integer power. The general form of a polynomial function in one variable, $x$, is given by $a_n{x}^n+a_{n-1}{x}^{n-1}+...+a_2{x}^2+a_{1}x+a_0$, where $a_n$, $a_{n-1}$,...,$a_0$ are constants, and $n$ is a non-negative integer.

For the function under consideration, $f(x)=\frac{4x^3+1}{2x^3+\sqrt{16x^6+1}}$, both the numerator and the denominator consist of polynomial expressions. The highest power of the variable $x$, known as the degree of the polynomial, plays a crucial role in determining the end behavior of the function. It's also used to determine the function's overall shape, symmetry, and the number of roots or x-intercepts it may have. These are foundational concepts that aid in understanding the more nuanced aspects of calculus.

Limits at Infinity

The concept of limits at infinity is concerned with the behavior of a function as the input grows without bound. In other words, it offers a way to describe the value that a function is approaching as $x$ becomes very large or very small (in the negative direction).

Taking limits at infinity helps to ascertain the horizontal asymptotes as we have done for our function $f(x)=\frac{4x^3+1}{2x^3+\sqrt{16x^6+1}}$. By dividing each term by the highest power of $x$ in the denominator, as shown in the steps, and then simplifying, we can isolate the terms that approach zero as $x$ approaches infinity. The limit of such expressions is generally found by noting that any fraction with $x$ in the denominator will approach zero as the value of $x$ gets arbitrarily large.

The conclusion that the limit of $f(x)$ as $x$ approaches infinity is zero is inferred by observing that after a certain point, changes in the input do not numerically affect the output; it will stay anchored at the horizontal asymptote. This insight is fundamental in understanding the long-term behavior of functions and is vital for fields ranging from physics to economics where such trends appear frequently.