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)=40x5+x216x4−2xf\left(x\right)=\frac{$$40x^5$+x^2$$}{$$16x^4-2x$$}

Accepted Answer

Identify the degrees of the polynomials in the numerator and the denominator. The numerator is $$40x^5$$ + $$x^2$$, which has a degree of 5, and the denominator is $$16x^4 - 2x$$, which has a degree of 4. Since the degree of the numerator (5) is greater than the degree of the denominator (4), the limit as x approaches infinity will be infinity. This means there is no horizontal asymptote as x approaches infinity. To find the limit as x approaches negative infinity, observe that the behavior will be similar to the limit as x approaches positive infinity because the highest degree terms dominate. Thus, the limit as x approaches negative infinity will also be infinity, indicating no horizontal asymptote in this direction either. For a more precise analysis, divide every term in the numerator and the denominator by $$x^5$$, the highest power of x in the numerator. This simplifies the expression to (40 + x^(-3))/(16x^(-1) - 2x^(-4)). As x approaches infinity or negative infinity, the terms with negative exponents approach zero, simplifying the expression to 40/0, which confirms that the limits are infinite and there are no horizontal asymptotes.

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).

$f (x) = \frac{40 x^{5} + x^{2}}{16 x^{4} - 2 x}$

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Key Concepts

Limits at Infinity

Horizontal Asymptotes

Rational Functions