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Calculus
Concept Version 6
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Infinite Limits

Limits involving infinity can be formally defined using a slight variation of the (ε,δ)(\varepsilon, \delta)(ε,δ)-definition.

Learning Objective

  • Evaluate the limits of functions as xxx approaches infinity


Key Points

    • For f(x)f(x)f(x) a real function, the limit of fff as xxx approaches infinity is LLL, means that for all ε>0\varepsilon > 0, there exists ccc such that whenever x>cx > cx>c, |f(x)−L|<ε|f(x) - L| < \varepsilon.
    • For a rational function f(x)f(x)f(x) of the form p(x)q(x)\frac{p(x)}{q(x)}​q(x)​​p(x)​​, there are three basic rules for evaluating limits at infinity, where p(x)p(x)p(x) and q(x)q(x)q(x) are polynomials.
    • If the limit at infinity exists, it represents a horizontal asymptote at y=Ly = Ly=L.

Terms

  • asymptote

    a straight line which a curve approaches arbitrarily closely, as they go to infinity

  • definition

    a formalization of the notion of the limit of functions


Full Text

Limits involving infinity can be formally defined using a slight variation of the (ε,δ)(\varepsilon, \delta)(ε,δ)-definition. For f(x)f(x)f(x) a real function, the limit of fff as xxx approaches infinity is LLL, denoted limx→∞f(x)=L\lim_{x \to \infty}f(x) = Llim​x→∞​​f(x)=L, means that for all ε>0\varepsilon > 0ε>0, there exists ccc such that ∣f(x)−L∣<ε\left | f(x) - L \right | < \varepsilon∣f(x)−L∣<ε whenever x>cx>cx>c. Or, formally:

∀ε>0∃c∀x<c:∣f(x)−L∣<ε\forall \varepsilon > 0 \; \exists c \; \forall x < c :\; \left | f(x) - L \right | < \varepsilon∀ε>0∃c∀x<c:∣f(x)−L∣<ε

Infinite Limit

For any arbitrarily small ε\varepsilonε, there exists a large enough NNN such that when x>Nx > Nx>N, ∣f(x)−2∣<ε\left | f(x)-2 \right | < \varepsilon∣f(x)−2∣<ε. Therefore, the limit of this function at infinity exists.

Similarly, the limit of fff as xxx approaches negative infinity is LLL, denoted limx→−∞f(x)=L\lim_{x \to -\infty}f(x) = Llim​x→−∞​​f(x)=L, means that for all ε>0\varepsilon > 0ε>0 there exists ccc such that ∣f(x)−L∣<ε|f(x) - L| < \varepsilon∣f(x)−L∣<ε whenever x<cx<cx<c.

For a rational function f(x)f(x)f(x) of the form p(x)q(x)\frac{p(x)}{q(x)}​q(x)​​p(x)​​, there are three basic rules for evaluating limits at infinity (p(x)p(x)p(x) and q(x)q(x)q(x) are polynomials):

  1. If the degree of ppp is greater than the degree of qqq, then the limit is positive or negative infinity depending on the signs of the leading coefficients;
  2. If the degree of ppp and qqq are equal, the limit is the leading coefficient of ppp divided by the leading coefficient of qqq;
  3. If the degree of ppp is less than the degree of qqq, the limit is 000.

If the limit at infinity exists, it represents a horizontal asymptote at y=Ly = Ly=L. Polynomials do not have horizontal asymptotes; such asymptotes may however occur with rational functions.

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