Determine lim⁡x→∞f(x)\$$lim_{x\rightarrow\infty}f$$\left(x\right)limx→∞f(x) and lim⁡x→−∞f(x)\$$lim_{x\rightarrow-\infty}f$$\left(x\right)limx→−∞f(x) for the following functions. Then give the horizontal asymptotes of fff (if any).f(x)=4x3+12x3+16x6+1f\left(x\right)=\frac{$$4x^3+1$$}{$$2x^3$$+\sqrt{$$16x^6+1$$}}

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.5.47

Chapter 2, Problem 2.5.47

Determine $$\lim$_{x$\rightarrow\]\infty$}f$\left$(x$\right$)$ and $$\lim$_{x$\rightarrow$-$\infty$}f$\left$(x$\right$)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f (x) = \frac{4 x^{3} + 1}{2 x^{3} + \sqrt{16 x^{6} + 1}}$

Verified step by step guidance

First, identify the highest power of x in both the numerator and the denominator. In the given function f(x) = $\frac{4x^3+1}{2x^3+\sqrt{16x^6+1}$}, the highest power of x in the numerator is x^3 and in the denominator is x^3 (from the term $\sqrt{16x^6+1}$).

To simplify the expression, divide every term in the numerator and the denominator by x^3, the highest power of x in the denominator.

After dividing, the function becomes f(x) = $\frac{4 + \frac{1}{x^3}$}{2 + $\sqrt{16 + \frac{1}{x^6}$}}.

Now, evaluate the limit as x approaches infinity. As x approaches infinity, the terms $\frac{1}{x^3}$ and $\frac{1}{x^6}$ approach 0. Thus, the function simplifies to $\frac{4}{2 + \sqrt{16}$}.

Similarly, evaluate the limit as x approaches negative infinity. The simplification process is the same, and the function again simplifies to $\frac{4}{2 + \sqrt{16}$}. Therefore, the horizontal asymptote is y = $\frac{4}{2 + \sqrt{16}$}.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits at Infinity

Limits at infinity describe the behavior of a function as the input approaches positive or negative infinity. This concept is crucial for understanding how functions behave in extreme cases, allowing us to determine horizontal asymptotes. For example, if the limit of f(x) as x approaches infinity exists and is a finite number, it indicates that the function approaches a horizontal line at that value.

Horizontal Asymptotes

Horizontal asymptotes are lines that a graph approaches as x approaches infinity or negative infinity. They provide insight into the end behavior of a function. If a function has a horizontal asymptote at y = L, it means that as x becomes very large or very small, the function values get closer to L, indicating stability in the function's output at extreme inputs.

Rational Functions

Rational functions are ratios of polynomials, expressed in the form f(x) = P(x)/Q(x), where P and Q are polynomials. The degrees of these polynomials significantly influence the limits at infinity and the existence of horizontal asymptotes. For instance, if the degrees of P and Q are equal, the horizontal asymptote can be found by taking the ratio of their leading coefficients.

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Related Practice

Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→1 x^4=1

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Textbook Question

A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2,4,6,8,…} is specified by the function f(n) = 2n, where n=1,2,3,….The limit of such a sequence is lim n→∞ f(n), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences or state that the limit does not exist.

{2,3/4,4/9,5/16,…}, which is defined by f(n) = (n+1) / n^2, for n=1,2,3,…

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Textbook Question

Use a graph of f to estimate $\lim_{x \to a} f (x)$ or to show that the limit does not exist. Evaluate f(x) near $x = a$ to support your conjecture.

$f (x) = \frac{1 - \cos (2 x - 2)}{{(x - 1)}^{2}}; a = 1$

Textbook Question

Consider the position function s(t)=−16t^2+128t (Exercise 13). Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=1. <IMAGE>

Textbook Question

Determine the points on the interval (0, 5) at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated. <IMAGE>

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Textbook Question

Find all vertical asymptotes $x = a$ of the following functions. For each value of $a$ , determine $\lim_{x \to a^{+}} f (x)$ , $\lim_{x \to a^{-}} f (x)$ , and $\lim_{x \to a} f (x)$ .

$f (x) = \frac{\cos (x)}{x^{2} + 2 x}$