Question

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)=x6+834x2+3x4+1f\left(x\right)=\frac{\sqrt[3]{$$x^6+8$$}}{$$4x^2$$+\sqrt{$$3x^4+1$$}}

Accepted Answer

First, identify the dominant terms in the numerator and the denominator as x approaches infinity or negative infinity. For the numerator $$ \sqrt[3]{x^6 + 8} $$, the dominant term is $$ x^2 $$ because $$ \sqrt[3]{x^6} = x^2 . $$For the denominator $$ 4x^2 + \sqrt{3x^4 + 1} $$, the dominant term is $$ \sqrt{3x^4} = \sqrt{3}x^2 . $$Simplify the expression by dividing both the numerator and the denominator by $$ x^2 $$, the highest power of x in the denominator. This gives $$ \frac{x^2}{x^2} = 1  in $$the numerator and $$ \frac{4x^2}{x^2} + \frac{\sqrt{3x^4}}{x^2} = 4 + \sqrt{3}  in $$the denominator. Now, evaluate the limit as $$ x 	o \infty . $$The expression simplifies to $$ \frac{1}{4 + \sqrt{3}} . $$Since the terms involving $$ x  in $$the numerator and denominator cancel out, this is the horizontal asymptote as $$ x 	o \infty . $$Evaluate the limit as $$ x 	o -\infty . $$The simplification process is similar, but consider the behavior of $$ x^2 $$ and $$ \sqrt{3}x^2  as$$ $$ x 	o -\infty . $$The expression remains $$ \frac{1}{4 + \sqrt{3}} $$, indicating the same horizontal asymptote. Conclude that the horizontal asymptote of $$ f(x)  is$$ $$ y = \frac{1}{4 + \sqrt{3}} $$ for both $$ x 	o \infty $$ and $$ x 	o -\infty .$$

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).

$f (x) = \frac{\sqrt[3]{x^{6} + 8}}{4 x^{2} + \sqrt{3 x^{4} + 1}}$

Verified video answer for a similar problem:

Key Concepts

Limits at Infinity

Horizontal Asymptotes

Rational Functions