Find each limit, if possible. (a) limx →∞(x^2+2)/(x^3-1) (b) limx →∞(x^2+2)/(x^2-1)

step 1 Hello friends, we have to find the limit x tends to infinity x square plus 2 upon x cube minus 1

Find each limit, if possible. (a) limx...

Find each limit, if possible.
$$
\begin{array}{l}{\text { (a) } \lim _{x \rightarrow \infty} \frac{x^{2}+2}{x^{3}-1}} \\ {\text { (b) } \lim _{x \rightarrow \infty} \frac{x^{2}+2}{x^{2}-1}} \\ {\text { (c) } \lim _{x \rightarrow \infty} \frac{x^{2}+2}{x-1}}\end{array}
$$

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Key Concepts

Limits at Infinity

This concept involves finding the behavior of a function as the variable grows without bound. In the context of rational functions, it often requires determining which terms dominate the behavior as the variable increases indefinitely.

Rational Functions

Rational functions are expressed as the quotient of two polynomials. Their limit behavior, especially as the variable approaches infinity, is governed by the relationship between the degrees of the numerator and denominator polynomials.

Degree Analysis

This key idea involves comparing the degrees (highest exponents) of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the limit at infinity is zero; if the degrees are equal, the limit is the ratio of the leading coefficients; and if the numerator’s degree exceeds that of the denominator, the function typically diverges.

Horizontal Asymptotes

Horizontal asymptotes represent the value that a function approaches as the independent variable tends toward plus or minus infinity. In rational functions, these are directly related to the degrees of the polynomial in the numerator and denominator, indicating the limit at infinity.

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