Determine the behaviour of the following functions as x →±∞: (a) f(x)=(x^3+x+1)/(2-3 x^3) (b) f(

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Determine the behaviour of the following functions as x →±∞:...

Determine the behaviour of the following functions as $x \rightarrow \pm \infty$ :
(a) $f(x)=\frac{x^{3}+x+1}{2-3 x^{3}}$
(b) $f(x)=\mathrm{e}^{-x} \sin x$
(c) $f(x)=[x] / x$, where $[x]$ denotes the integer part of $x$

Key Concepts

Limits at Infinity

This is the evaluation of the behavior of functions as the variable approaches positive or negative infinity. It focuses on determining the end behavior of functions and involves identifying horizontal asymptotes or unbounded behavior by analyzing the dominant terms of the expressions.

Dominant Term Analysis in Rational Functions

When dealing with rational functions—the quotient of two polynomials—the asymptotic behavior is largely determined by the highest degree terms in the numerator and denominator. This technique simplifies limit calculations at infinity by reducing the expression to a comparison of these dominant terms.

Exponential Decay and Oscillatory Functions

In functions that combine exponential decay with oscillatory behavior, such as the product of an exponential function and a sine function, the exponential factor typically dictates the long-term behavior. Exponential decay can dampen oscillations, leading the overall function to tend towards zero despite the periodic fluctuations of the trigonometric component.

Properties of the Floor (Greatest Integer) Function

The floor function maps a real number to the greatest integer less than or equal to it. Analyzing its behavior within ratios or limits requires careful attention to its step-wise nature and discontinuities, particularly when it is divided by a continuous variable, which affects how the quotient behaves as the variable approaches infinity.

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